Biological discovery using gene regulatory networks generated from multiple-disruption expression libraries

ABSTRACT

Embodiments of this invention include application of new inferential methods to analysis of complex biological information, including gene networks. In some embodiments, disruptant data and/or drug induction/inhibition data are obtained simultaneously for a number of genes in an organism. New methods include modifications of Boolean inferential methods and application of those methods to determining relationships between expressed genes in organisms. Additional new methods include modifications of Bayesian inferential methods and application of those methods to determining cause and effect relationships between expressed genes, and in some embodiments, for determining upstream effectors of regulated genes. Additional modifications of Bayesian methods include use of heterogeneous variance and different curve fitting methods, including spline functions, to improve estimation of graphs of networks of expressed genes. Other embodiments include the use of bootstrapping methods and determination of edge effects to more accurately provide network information between expressed genes. Methods of this invention were validated using information obtained from prior studies, as well as from newly carried out studies of gene expression.

RELATED APPLICATIONS

[0001] This application claims priority under 35 U.S.C. Section 119(e) to U.S. provisional applications serial No. 60/325,016, filed Sep. 26, 2001; No. 60/334,372, filed Nov. 29, 2001; No. 60/334,255, filed Nov. 29, 2001; No. 60/334,230, filed Nov. 29, 2001; No. 60/370,824, filed Apr. 8, 2002; and No. 60/397,458 filed Jul. 19, 2002. Each of the above provisional patent applications is herein incorporated fully by reference.

FIELD OF THE INVENTION

[0002] Embodiments of this invention relate to methods for elucidating network relationships between genes. Methods include Boolean logic, Bayesian estimation, maximum likelihood analysis, spline functions and other curve-fitting methods, and methods to determine edge relationships between groups of genes that are functionally related to each other.

BACKGROUND

[0003] Gene regulatory networks elucidated from strategic, genome-wide experimental data can aid in the discovery of novel gene function information and expression regulation events from observation of transcriptional regulation events among genes of known and unknown biological function.

[0004] Methods designed to elucidate gene regulation pathways have been reported previously^((1,2,3)). The inferred networks reported in these studies were derived from gene expression data sets derived from time course, cell cycle and environmental perturbation^((4,5)). However, control relationships inferred from such data sets are suspect since they are not based on comprehensive experimental data designed to elucidate transcription-related regulatory control functions. To rigorously and precisely identify novel and complex gene regulatory networks from de novo expression data sets, a systematic and integrated strategy of expression experiments on genomic deletion mutants combined with suitable computational methods is necessary ⁽⁶⁻⁹⁾.

[0005] Of particular importance in the creation of inferred regulatory networks is the biological relevance of the expression experiments from which the putative control relationships are derived. Genome-wide expression data generated from competitive hybridization disruption experiments offer the advantages of internal control and quantitative measurement of the direct effects of a gene's presence or absence on the expression of other genes. Selection of disruptant experiments to maximize the elucidation of control relationships is valuable for generating useful gene regulatory information.

SUMMARY

[0006] The present invention provides methods useful for establishing interrelationships among genes or groups of genes in a system, e.g., establishing gene pathways or gene networks in a system. In one embodiment, gene networks or gene pathways can be constructed based on analyzing gene disruption expression profiles using improved Boolean methods, improved Bayesian methods, or a combination of Boolean and Bayesian methods, as provided by the present invention. In another embodiment, gene networks or gene pathways can be constructed based on analyzing expression profiles of genes affected by an agent, e.g., a drug. In yet another embodiment, gene networks or gene pathways affected by an agent can be constructed by analyzing the gene networks or pathways obtained from the gene disruption expression profiles and the agent affected expression profiles.

[0007] In general, gene disruption expression profiles can be obtained based on expression profiles of a library of genes, each of which disrupted individually or in combination with others, e.g., other genes in a related function to provide an expression profile. For example, a library of genes can be selected (e.g., an entire genome or a series of genes selected based on other criteria, including polymerase chain reaction (PCR) primers). Regardless of how the library of genes is selected, once selected, each gene of the library can be disrupted individually and/or in combination with others resulting in a collection of libraries, each of which containing at least one disrupted gene along with other non-disrupted genes. Thus, if one selects 100 genes, the resulting library will consist of at least 101 different sub-libraries e.g., one sub-library of non-disrupted genes or wild-type and at least one disruptant sub-library for each of the 100 genes. Thus, libraries of disrupted genes can be created and expression profiles for each sub-library and the entire library can be obtained.

[0008] Agent affected expression profiles can be obtained by administering one or more desired agents to a system containing selected genes and collecting expression profiles of the genes at different dosage or time point of agent administration. In some cases the agent has no effect on gene expression, in other cases the agent is inhibitory (e.g., decreases gene expression) and in other cases the agent increases gene expression (e.g., is an inducer).

[0009] Gene expression profiles can be obtained in a quantified form using any suitable means, e.g., using microarray. In one embodiment, a gene expression profile can be organized into gene expression matrix, e.g., a binary matrix categorizing the genes into affected and not affected. In another embodiment, one can introduce an “equivalence set” in which data is normalized, thereby revealing the quantitative relationships in gene expression. From an equivalence set, network information relating the genes with each other can be created. Then, one can use networks to determine functional relationships between genes. One can then use the deduced functional relationships between genes to predict drug and or biological effects. Then, those predictions can be experimentally tested using, for example, mircoarray experiments. This process can result in what we call a “final common pathway” of gene expression, that is, alteration in the function of one gene results in effects on genes “downstream” from the altered gene.

[0010] According to one embodiment of the present invention, one can analyzing “downstream” effect from a directly affected gene by using the improved Boolean methods of the present invention. According to another embodiment of the present invention, one can go “upstream” from an affected gene by applying an improved Bayesian method of the present invention. The present invention provides a new approach to Bayesian gene network analysis, in which non-linear, and/or non-parametric regression models are used. Using this approach means that one need not make any apriori assumptions about causal relationships, but rather can infer causal relationships, thereby providing “upstream” or “initial pathways” by which a gene observed to be affected by a drug or treatment is induced to alter its expression by other genes in an upstream relationship to the affected, observed gene. The present invention provides an improved criterion, herein termed “BNRC” for estimating a Bayesian graph of gene networks. Thus, the gene relationships are selected that minimize the BNRC criterion.

[0011] In other embodiments, other new methods are provided that not dependent upon Gaussian or other assumed variance in the data. Rather, in certain embodiments, we can measure the variance of the data and use the observed variance to affect the BNRC criterion. Using such non-parametric regression with heterogeneous error variances and interactions, one can optimize the curve fitting of data obtained, and predict how many experiments are needed to obtain data having a desired variance. Using methods such as these, one can obtain gene network information that has a desired degree of accuracy and reliability, and provides both upstream and downstream effects. In some embodiments, the new Bayesian model and penalized maximum likelihood estimation (PMLE) yield similar results.

[0012] In still other embodiments, relationships between two genes can be elucidated by analysis of a graph of expression of one gene compared to the expression of another gene. Such graphs may be linear or non-linear. To characterize the relationships, in some embodiments, linear splines, B-splines, Fourier transforms, wavelet transforms, or other basis functions can be used. In some cases, B-splines are conveniently employed.

[0013] Determining whether a particular gene or series of genes is important in regulating other genes can be very useful. However, in certain cases, outliers in the data can complicate interpretation of results. This problem can be especially troubling where outliers are near boundaries of gene groups in the overall network. Thus, in certain embodiments, boundary effects can be elucidated, in which the edge intensity and the degree of confidence of the direction of Bayes causality can be determined.

[0014] In other embodiments, time-course of gene expression can be determined using linear splines. Using splines, time-ordered data can be more reliably analyzed that prior art, “fold-change” methods, which can be relatively insensitive.

[0015] Using one or more of the above-described general methods, the present invention provides gene network information that is useful and validated by certain results already known for the yeast genome. However, the methods are widely applicable to any genetic network (e.g., “transcriptome”), and to interactions at the level of proteins (“proteome”). Additionally, using one or more of the new methods, we have determined novel relationships between functional genes relating to antifungal therapies. Thus, using methods of this invention, one can predict putative therapeutic targets based on an understanding of the initial and final common pathways of gene expression.

BRIEF DESCRIPTION OF THE DRAWINGS

[0016] This invention is described with respect to specific embodiments thereof, in which:

[0017]FIG. 1 depicts a schematic diagram of model-based interactive biological discovery of this invention.

[0018]FIG. 2 depicts construction of a gene regulatory sub-network model, using a Boolean method of this invention. FIG. 2A depicts a Gene Expression Matrix, in which numerous profiles are integrated. Each element of the matrix represents a ratio of gene expression between a gene in a column and a gene in a row. FIG. 2B depicts binary relationships between genes to produce a Binary Matrix. If gene ‘G1’ is deleted and the intensity of gene ‘G2’ is significantly altered as a result, then gene ‘G1’ affects gene ‘G2’. FIG. 2C depicts an adjacency matrix of identified looped regulation genes. If gene ‘G3’ and gene ‘G4’ affect each other mutually, they form a loop (strongly connected component) regulation. FIG. 2D depicts a step in which genes are partitioned into an Equivalence Set, which treats a loop logically as a “virtual gene”. An equivalence set is a group of genes that affect each other or as a group affect one discrete gene. FIG. 2E depicts a skeleton relation between virtual genes. The shortest path relationships should be selected to build hierarchical connections. FIG. 2F depicts a regulation pathway formed from the skeleton matrix.

[0019]FIG. 3 depicts results of a disruptant-based study and analysis of this invention in yeast in which 552 nodes representing the included genes and 2953 putative regulatory links among these genes.

[0020]FIG. 4 depicts a transcription factor regulatory network model of this invention classified by cellular functional roles (CFR) in yeast. In this model, there were 98 transcription factors grouped by cellular functional roles according to information provided in the Yeast Proteome Database. Genes inside the circles are grouped into a given cellular functional category. Regulatory control relationships are depicted with colored lines. Colors indicate the category of genes from which the control relationships emanate. Blue: Carbohydrate metabolism, Bluish purple: Chromatin/Chromosome structure, Brown: Energy generation, Dark Green: Other metabolism, Gray: DNA repair, Green: Lipid, fatty acid metabolism, Light Green: Amino acid metabolism, Orange: Cell stress, Red: Meiosis/Mating response, Pink: Differentiation, Purple: Cell cycle. The bold red lines indicate regulatory control of transcription factors related to cell cycle originating from genes in the meiosis/mating response group. The bold blue lines indicate control relationships emanating from the carbohydrate metabolism group exerting influence over genes in the lipid fatty-acid metabolism group. Several control lines emanating from “carbohydrate metabolism” are depicted. In sharp contrast, only 2 individual genes, SKN7 and HMS2 independently influence cell cycle related gene expression.

[0021]FIG. 5 depicts a detailed view of a gene regulatory sub-network of transcription factors reconstructed from Boolean analysis of gene expression experiments on disruptant mutant yeast strains. Black lines indicate a regulatory relationship, with the arrows showing the direction of expression influence. The colors and shapes of the nodes denote the general categories of cellular function of the gene product according to their descriptions in the YPD. Genes related to cell division mechanisms are indicated with triangular nodes and genes related to DNA repair and chromosome structure are depicted with squares. The elucidated network shows novel topological control relationships among genes related to meiosis, the mating response and DNA structure and repair mechanisms via UME6 and MET28. Genes related to meiosis and mating response genes are downstream of a cascade regulated by INO2 and a further sub-grouping of genes related to DNA repair and structure appears hierarchically downstream of MET28.

[0022]FIG. 6 depicts a flow chart showing methods of this invention for the estimation of network relationships between genes using Bayesian inference and minimization of a BNRC criterion.

[0023]FIGS. 7a and 7 b depict conventional methods for analyzing drug response expression data. FIG. 7a includes list of Yeast genes whose expression levels are significantly affected by Griseofulvin exposure at 100 mg at 1 minute (one of 20 experiments). FIG. 7b depicts hierarchical clustering of expression data from drug response and gene disruption experiments.

DETAILED DESCRIPTION

[0024] The descriptions that follow include specific examples drawn from studies in the yeast, Sarchomyces. The methods for analyzing relationships between yeast genes are equally applicable to analysis of relationships among genes of different species, including eukaryotes, prokaryotes, and viruses. Thus, the descriptions and examples that follow are illustrative and are not intended to limit the scope of the invention.

[0025] 1. Biological Discovery with Gene Regulatory Networks in Yeast Generated from Multiple-Disruption Full Genome Expression Libraries

[0026] Gene regulatory networks elucidated from strategic, genome-wide experimental data can aid in the discovery of novel gene function information and expression regulation events form observation of transcriptional regulation events among genes of known and unknown biological function. Of particular importance in the creation of inferred regulatory networks is the biological relevance of the expression experiments from which the putative control relationships are derived. Genome-wide expression data generated from competitive hybridization disruption experiments offer the advantages of internal control and quantitative measurement of the direct effects of a gene's presence or absence on the expression of other genes. Selection of disruptant experiments to maximize the elucidation of control relationships is important for generating useful gene regulatory information. To create a reliable and comprehensive data set for the elucidation of transcription regulation on 120 genes with known involvement in transcription regulation. We report several novel regulatory relationships between known transcription factors and other genes with previously unknown biological function discovered with this expression library.

[0027] We have implemented, for the yeast genome, a systematic, iterative approach that combines full-genome biological expression experiments with gene regulatory inference, as shown in FIG. 1. In certain embodiments, we start with the creation of a library of hundreds of full genome expression experiments, with one gene's expression disrupted per experiment. From this data we used computational techniques to infer approximations of gene expression regulatory relationships. We then examined the biological relevance of regulatory relationships with computerized visualization and simulation software and validated our findings on novel or biologically interesting sub-networks through other databases as well as through further experimentation, including combinatorial disruption experiments.

[0028] We constructed a gene expression data library using full-genome yeast c-DNA microarrays. The library was comprised of expression experiments on 120 yeast strains, each with one gene disrupted by homologous recombination. Each of the genes in this library was selected for experimentation because it was reported in the Yeast Proteome Database (YPD) to be a factor involved in transcription regulation. Previously reported gene regulatory networks show that genes can interact with themselves and with other regulatory genes⁽¹⁰⁾. To reconstruct hierarchical regulatory relationships from the expression library, in certain embodiments, we developed a novel Boolean algorithm that accommodates common looped regulatory relationships. As shown in FIG. 2, gene regulatory relationships modeled by this method can be represented as a directed graph of up-regulation or down-regulation of gene expression between 2 given genes of the 5871 genes measured in the each expression experiments. We constructed a 552 genes member model of regulatory control relationships and then further constrained a sub-network model composed of 98 well known transcription factors. The resultant model, shown in FIG. 3, contains a total of 552 nodes representing the included genes and 2953 putative regulatory links among these genes.

[0029] In certain embodiments, we classified transcription factors in the network model according to cellular functional roles (CFR) as defined in YPD. FIG. 4 shows the control relationships among classified transcription factors in the network. Only SKN7 and HMS2 directly influenced expression of cell cycle related genes. We identified several control lines emanating from “carbohydrate metabolism” genes to all other functional gene groups. This finding is consistent with the energy dependent nature of many cellular processes and metabolic pathways.

[0030] As shown in FIG. 4, a distinct feature is that expression levels of lipid fatty-acid metabolism transcription factors were exclusively under control of carbohydrate metabolism transcription factors. This relation was given an account of interactions among proteins involved in phospholipid synthesis pathway with the glucose response pathway, the lipid signaling pathway and other lipid synthesis pathways have been reported⁽¹¹⁾.

[0031] Our new methods were employed to further explore detailed relationships between expression regulatory genes, such as transcription factors with regulatory and non-regulatory genes, from all of gene expression experimental data. One can characterize regulatory roles of genes with unreported biological function by virtue of their expression control by and/or over genes with known function. FIG. 5 shows an example of newly elucidated control relationships among transcription factors involved in cell division regulation and DNA replication/repair regulation. We found two discrete functional branches in the sub-network that correspond to cell division regulation and DNA replication/repair are linked by UME6 and MET28, indicating the important role of these two transcription factors in coordinating expression regulation of these interdependent regulatory pathways. MET28, as its name suggests, was previously characterized as a transcription factor related to methionine metabolism⁽¹²⁾. The novel putative role for Met28p in regulation of chromosome segregation is supported by its reported interaction with known chromosomal segregation component Smc1p, as part of a larger nexus of chromosomal segregation proteins, in mating-type two-hybrid assays⁽¹³⁾.

[0032] Through sequence analysis of coding sequences and upstream regions of genes in the above-mentioned sub network, we validated the sequence level control mechanisms between transcription factors and their target genes and DNA binding sequences. In the case of UME6, which is known as a global transcriptional regulation of many meiotic genes ^((14,15)) and its control system, we performed Multiple Expectation-Maximization for motif Elicitation (MEME) analysis of regions upstream of the 34 genes controlled by UME6 in our model ⁽¹⁶⁾. We found two consensus sequences, TAGCCGCCGA (SEQ ID NO: 1) and TGGGCGGCTA (SEQ ID NO: 2) that were present in 14.7% and 32.4% of the 34 genes respectively, and that had significant P values according to the MEME search. According to the TRANSFAC database, TAGCCGCCGA is defined as the binding site of Ume6p ⁽¹⁴⁾ and TCGGCGGCA is reported to be the binding site of a repressor of CAR1 ⁽¹⁷⁾ which were repressed by a three components complex containing Ume6p (Ume6p, Sin3p and Rpd3p) ⁽¹⁸⁾. Aside from the Ume6p related binding motifs, no other MEME consensus sequence was present upstream of the 11 meiosis related genes, a finding which suggests that these 11 genes are regulated exclusively by UME6 and that Ume6p directly influences their expression. Only two other genes possessed the putative binding sequence but did not show expression influence on the count of UME6 in our experiments.

[0033] Experimentally driven discovery of network models of expression control allows for specific biological insights relevant to gene regulatory pathways that are not readily reconstructed from the available biological literature or present in pre-compiled pathway databases. We have shown here that such a system is useful in discovering novel gene function information as well as novel regulatory mechanisms. Use of this and similar strategies to elucidate hierarchical regulatory pathways from full genome expression libraries will allow for rapid insight into transcription regulation that can be applied to rational drug discovery and agrochemical targeting.

[0034] Methods

[0035] Boolean Network Inference Algorithms

[0036] A gene expression matrix E is created from a set of gene disruption experiments. The value of matrix element E(a,b) indicates the expression ratio of gene ‘b’ to the normal condition in the disruptant in which gene ‘a’ which is caused by the deletion of gene ‘a’.

[0037] (1) Using the gene expression matrix E, if the intensity of gene ‘b’ is changed higher than a given threshold value θ, or is changed lower than a given threshold 1/θ resulting from the disruption of gene ‘a’, it is defined that gene ‘a’ affects gene ‘b’ in directly or indirectly and the value of element (a,b) in the binary matrix R is set to 1; R(a,b)=1. Thus the binary matrix R is created by cutting the value of each element in the gene expression matrix E at the threshold (θ or 1/θ).

[0038] (2) In the binary matrix R, if there is the relation that gene ‘a’ and ‘b’ affect each other, that is R(a,b)=R(b,a)=1, we cannot decide which gene is located at the upper stream. This is the limitation or disadvantage of this method, however, we introduce an equivalence set, which makes a set of the group consisting of genes affecting each other and the group is assumed to be one gene.

[0039] (3) Ordering genes (topological sort) Equivalence sets have the semi-order relation and we can be drawn up equivalence set in semi-order (topological sort) to infer the network

[0040] (4) Skeleton matrix; semi-ordered accessibility matrix between equivalence sets includes indirect affections. In order to remove them and to make a skeleton matrix, we set the rank to each equivalence set defined as follows: Equivalence set belonging to rank 1 gives no indirect affection to another equivalence sets. Equivalence set belonging to rank 3 gives direct affection to the sets with rank 2 and does indirect affection to ones with rank 1. After setting the rank to each equivalence set, remove all indirect affection from the semi-ordered accessibility matrix.

[0041] (5) Draw multi-level digraph; draw the lines between nodes based on the value of each element in the skeleton matrix.

[0042] Microarray Experiments

[0043] Information on gene expression can be obtained using any conventional methods known in the art. For example, microarrays can be used in which complementary DNAs (cDNA) unique to each gene whose expression is to be measured are placed on a substrate. RNA obtained from samples can be analyzed by hybridization to corresponding cDNA, and can be detected using a variety of methods. In some embodiments, it can be desirable to use a microarray detectors and/or methods as described in U.S. Provisional Patent Application Serial No. 60/382,669, filed May 23, 2002, herein incorporated fully by reference.

[0044] We collected gene expression data using full-genome yeast c-DNA microarrays⁽¹³⁾ BY4741 (MATa, HIS3D1, LEU2D0, MET15D0, URA3D0) served as the wild type strain. Gene disruptions for strain BY4741 were purchased from Research Genetics, Inc. Cells were inoculated and grown in YPD (1% yeast extract, 1% bacto-peptone, 2% glucose) at 30° C. until OD₆₀₀ reached 1.0 in the logarithmic growth phase and harvested to isolate mRNA for assay of gene expression. The parental strain was the control used for each disruptant strain.

[0045] Data Normalization

[0046] We measured the quantities of 5871 mRNA species from 155 disruptants by cDNA microarray assay. A difference in fluorescent strength between Cy3, Cy5 causes bias of the expression quantity ratio. We normalized the expression quantity ratios of each expression profile. The ratio bias had a fixed trend in each spotted block, thus we calculated a linear regression to normalize the mean value ratio of each block to 1.0. The logarithm value of the ratio was used to indicate the standard expression level, therefore we found the logarithm value of ratio and calculated the average and standard deviation of these log values (see table1). The Standard Deviation (SD) of expression levels of all spotted genes from the UME6 (YDR207C) disruptant expression array for which UME6 is defined as a “Global Regulator” in YPD⁽³³⁾ disruptant was 0.4931, therefore we recognize that there may be an unacceptable number of errors in array data whose overall SD was larger than 0.5. Thus, in certain embodiments, one can eliminate such experiments from analysis. It can be appreciated that one can select the SD of the data in different ways. By selecting a SD of less than 0.5, one can take a relatively conservative approach, in that errors of the above type are relatively unlikely to affect the overall gene network inference. However, one can appreciate that SD of less than 0.4, less than 0.3, less than 0.2, less than 0.1, less than about 0.05, less than about 0.01, less than about 0.005, or less than about 0.001 can be selected.

[0047] Selection of Genes

[0048] In YPD, 314 genes were defined as “Transcription Factors”, and 98 of these have previously been studied for control mechanism. The expression profile data of 552 genes including the genes that controlled by this 98 “Transcription factors” were selected from 5871 profiles. Thus we constructed the gene regulatory network from the expression profile data set based on the values of these 552 genes in 120 gene disruption experiments.

REFERENCES

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[0067] II. Estimation of Genetic Networks and Functional Structures Between Genes Using Bayesian Networks and Nonparametric Regression

[0068] 1. Introduction

[0069] Improvement of the genetic engineering provides us enormous amount of valuable data such as microarray gene expression data. The analysis of the relationship among genes has drawn remarkable attention in the field of molecular biology and Bioinformatics. However, due to the cause of dimensionality and complexity of the data, it will be no easy task to find structures, which are buried in noise. To extract the effective information from biological data, thus, theory and methodology are expected to be developed from a statistical point of view. Our purpose is to establish a new method for understanding the relationships among genes clearer. Example 1 below provides the mathematical basis for these methods.

[0070] In certain embodiments of this invention, Bayesian networks are adopted for constructing genetic networks using microarray gene expression data from a graph-theoretic approach. Friedman and Goldszmidt (1998) proposed an interesting method for constructing genetic links by using Bayesian networks. They discretized the expression value of genes and considered to fit the models based on multinomial distributions. However, a problem still remains in choosing the threshold value for discretizing (by not only the experiments). The threshold value assuredly gives essential changes of affects of the results and unsuitable threshold value leads to incorrect results. On the other hand, recently, Friedman et al (2000) pointed out that discretizing could result in losses of information. To use the expression data as continuous values, they used Gaussian models based on linear regression. However this model can only detect linear dependencies and cannot produce a complete picture of relationships. We developed a new method for constructing genetic networks using Bayesian networks. To capture not only linear dependencies but also nonlinear structures between genes we use nonparametric regression models with Gaussian noise. Nonparametric regression has been developed in order to explore the complex nonlinear form of the expected responses without the knowledge about the functional relationship in advance. Due to the new structure of the Bayesian networks, a suitable criterion was needed for evaluating models. Thus, this invention includes a new criterion from Bayesian statistics. By using these methods we overcame defects of previous methods and attained more information. In addition, our methods include previous methods as special cases. We validated our methods through the analysis of the S. cerevisiae cell cycle data.

[0071] We found that using Bayesian analysis according to these methods, we could identify upstream causative relationships between genes, thereby identifying potential therapeutic targets.

[0072] 2. Bayesian Networks and Nonparametric Regression

[0073] Using a Bayesian network framework, we consider a gene as a random variable and decompose the joint probability into the product of conditional probabilities. For example, if we have a series of observations of the random vector, we can denote the probability of obtaining a given observation can depend upon the conditional probability densities. In certain embodiments, one can use nonparametric regression models for capturing the relationships between the variables. A variety of graphic tools can be used to elucidate the relationships. For example, polynomials, Fourier series, regression spline gases, B-spline bases, wavelet bases and the like can be used for defining a graph of gene relationships. One difficulty in selecting a proper graph is to properly evaluate variance and noise in the system.

[0074] 3. Criterion for Choosing a Proper Graph

[0075] In certain embodiments, we can let π(θ_(G) |λ) be the prior distribution on the unknown parameter θ_(G) with hyper parameter vector and let log π(θ_(G) |λ)=0(n). The marginal probability of the data X_(n) is obtained by integrating over the parameter space, and we choose a graph G with the largest posterior probability. Friedman and Goldszmidt (1998) considered the multinomial distribution as the Bayesian network model and also supposed the Dirichlet prior on the parameter θ_(G). In this case, the Dirichlet prior is the conjugate prior and the posterior distribution belongs to the same class of distribution. Then a closed form solution of the integration in (4) is obtained, and they called it BDe score for choosing graph. A BDe score is confined to the multinomial model, and we propose a criterion for choosing graph in more general and various situations.

[0076] A problem of constructing criteria based on the above model is how to compute the integration. Some methods that can be used include Markov chain, and Monte Carlo methods. In certain embodiments of this invention, we use the Laplace approximation for integrals. Thus, the optimal graph is chosen such that the criterion BNRC in Example 1 equation (5) is minimized.

[0077] This criterion is derived under log (θ_(G) |λ)=0(n). If log (θ_(G) |λ)=0(1), the mode θ_(G) is equivalent to the maximum likelihood estimate, MLE, and the criterion resulted in a Bayesian information criterion, known as BIC by removing the higher order terms 0 (n^(−j))(j≧0). Konishi (2000) provided a general framework for constructing model selection criteria based on the Kullback-Leibler information and Bayes approach. As can be seen from Example 1 below, the final graph can be selected as a minimizer of the BNRC and does not have to minimize each local score, BNRC_(j), because the graph is constructed as acyclic.

[0078] 4. Estimating Graphs and Related Structures Using a BNRC Criterion

[0079] For example, one can express our method in an illustration. Essential points of our method are the use of the nonparametric regression and the new criterion for choosing graph from Bayesian statistics. As for nonparametric regression in Section 2 of the this Example 1, we use B-splines as the basis functions. FIG. 1 of Example 1 is an example of B-splines of degree 3 with equidistance knots t₁ . . . , t₁₀. By using a backfitting algorithm, the modes can be obtained when the values of β_(jk) are given. The backfitting algorithm is shown in Example 1 below.

[0080] The modes depend on the hyper parameters β_(jk), and we have to choose the optimal value of β_(jk). In the context of our method the optimal values of β_(jk) are chosen as the minimizers of BNRC_(j).

[0081] The B-splines coefficient vectors can be estimated by maximizing equation (6) of Example 1. The modes of equation (6) are the same as the penalized likelihood estimates and we can look upon the hyper parameters λ_(jk) or β_(jk) as the smoothing parameters in penalized likelihood. Hence, the hyper parameters play an important role for fitting the curve to the data.

[0082] 5. Computation Experiments

[0083] We used Monte Carlo simulations to examine the properties of our method. Data were generated from an artificial graph and structures between variables (FIG. 2 of Example 1) and then the results summarized as follows: The criterion BNRC can detect linear and nonlinear structure of the data. A BNRC score may have a tendency toward over growth of the graph. Therefore, we use Akaike's information criterion, known as AIC and use both methods. AIC was originally introduced as a criterion for evaluating models estimated by maximum likelihood methods. But estimates by this method are the same as the maximum penalized likelihood estimate and is not MLE. The trace of S_(jk) shows the degree of freedom of the fitted curve and is a great help. That is to say, if trS_(jk) is nearly 2, the dependency can be considered linear. We use both BNRC and AIC for deciding whether to add a parent variable.

[0084] To validate these methods, we analyzed the S. cerevisiae cell cycle data discussed by Spellman et al. and Friedman et al. The data were collected from 800 genes and 77 experiments. We set the prior probability π_(G) as constant, because we have no information about the size of the true graph. The nonparametric regressors are constructed of 20 B-splines. The number of B-splines can be varied as desired. The hyperparameters control the smoothness of fitted curve and it can be desirable to select hyper parameters and the number of B-splines to provide a good fit to the data with a minimum of B-splines.

[0085] The results of the analysis can be summarized as follows: FIG. 3 of Example 1 shows BNRC scores when we predicted CLN2, CDC5 and SVS1 by one gene. The genes which give smaller BRNC scores produce a more reliable effect on the target gene. We can observe that which gene is associated with the target gene and we find the gene set which strongly depend on the expression of the target gene. In face, we can construct a brief network by using these information. We can look upon the optimal graph as a revised version of the brief network by taking account of the effect of interactions. If there is a linear dependency between genes, the score BNRC is good when the parent-child relationship is reversed. Therefore, the directions of the causal associations in the graph are not strict, especially when the dependencies are almost linear. There are some genes that mediate Friedman et al.'s result, such as MCD1, CSI2, YOX1 and so on. Most of these genes have been reported to play an important role. A large number of the relationships between genes are nearly linear. But we could find some nonlinear dependencies which linear models can rarely find.

[0086]FIG. 5 of Example 1 shows the estimated graph associated with genes which were classified by their processes into cell cycle and their neighborhood. We omitted some branches in FIG. 5, but important information is shown. As for the networks given by us and Friedman et al, we confirmed parent-children relationships and observed that both two networks are similar to each other. Especially, our network includes typical relationships which were reported by Friedman et al. As for the differences between both networks, we attention the parents of SVS1. Friedman et al. employed CLN2 and CDC5 as the parent genes of SVS1. On one hand, our result gives CSI2 and YKR090W for SVS1. Our candidate parent genes were found to be more appropriate than those of Friedman et al. Our model suitably fits to both cases in FIG. 4 of Example 1. Especially, we conclude that CDC5 has weak effects on SVS1 compared with other genes in Spellman's data (see also FIG. 3 of Example 1) In fact, as the parent gene of SVS1, the order of BNRC score of CDC5 is 247^(th). Considering the circumstances mentioned above, our method can provide us valuable information in understandable and useful form.

[0087]FIG. 6 schematically depicts a method of this invention for determining gene network relationships.

[0088] 6. Discussion

[0089] The new methods for estimating genetic networks from microarray gene expression data by using Bayesian network and nonparametric regression provide improved accuracy. We derived a new criterion for choosing graph theoretically, and represented its effectiveness through the analysis of the cell cycle data. The advantages of our methods include (1) we can use the expression data as continuous values; (2) we can also detect nonlinear structures and can visualize their functional structures being easily understandable; and (3) automatic search can accomplish the creation of an optimal graph.

[0090] Friedman et al's method retains known parameters such as threshold value for discretizing and hyper parameters in the Dirichlet priors selected by trial and error and were not optimized in a narrow sense. On the other hand, our method can automatically and appropriately estimate any parameter based on criterion which has sound theoretical basis.

[0091] In other embodiments we can derive criterion BNRC in more general situations. By way of example, we can construct graph selection criterion based on other statistical models.

[0092] III. Nonlinear Modeling of Genetic Network by Bayesian Network and Nonparametric Regression with Heterogeneous Error Variances and Interactions

[0093] In other embodiments, one can use different statistical methods for constructing genetic networks from microarray gene expression data based on Bayesian networks. In these embodiments, we estimate the conditional distribution of each random variable. We consider fitting the nonparametric regression models with heterogeneous error variances and interactions for capturing the nonlinear structures between genes. Once we set a graph using Bayesian network and nonparametric regression, a problem still remain to be solved in selecting an optimal graph, which gives a best representation of the system among genes. A new criterion for choosing graph from Bayes approach contains the previous methods for estimating networks using Bayesian methods. We demonstrated the effectiveness of proposed method through the analysis of Saccharomyces cerevisiae gene expression data newly obtained by disrupting 100 genes.

[0094] 1. Introduction

[0095] Use of Bayesian networks can be effective methods in modeling the phenomena through the joint distribution of a large number of variables. Friedman and Goldszmidt discretized the expression values and assumed the multinomial distributions as candidate statistical models. Peter et al. investigated the threshold value for discretizing. Friedman et al. pointed out that discretizing can lose some information considered to fit the linear regression models, which analyze the data as continuous. However, the assumption that the parent genes depend linearly on the objective gene is not always warranted. Imoto et al described the use of nonparametric additive regression models for capturing not only linear dependencies but also nonlinear relationships between genes.

[0096] In certain embodiments, we use Bayesian networks and the nonparametric heteroscedastic regression which can be more resistant to effect of outliers and can also capture effects of the interactions of parent genes.

[0097] After setting the graph, we evaluate its goodness or closeness to the true graph, which is completely unknown. The construction of a suitable criterion becomes important. Friedman and Goldszmidt derived the criterion, BDe, for choosing a graph based on multinomial models and Dirichlet priors. However, unknown hyper parameters in Dirichlet priors remain and one can only set the values experientially. We derived a new criterion for choosing a graph from Bayes approach. The criterion automatically optimizes the all parameters model and gives the optimal graph. In addition, our method includes the previous methods for constructing genetic network by Bayesian network. To show the effectiveness of the new method we analyze gene expression data of Saccharomyces cerevisiae by disrupting 100 genes.

[0098] 2. Bayesian Network and Nonparametric Heteroscedastic Regression Model with Interactions

[0099] Suppose that we have n sets of data {x₁, . . . ,x_(n)} of the p-dimensional random variable vector X=(X₁ . . . , X_(p))^(T) where x_(i)=x_(i1), . . . x_(ip))^(T) and x^(T) denotes the transpose of x. In microarray gene expression data, n and p correspond to the numbers of arrays and genes. Under Bayesian network framework, we considered a directed acyclic graph G and Markov assumption between nodes. The joint density function is then decomposed into the conditional density. Assuming that the conditional densities are parameterized by the parameter vectors θ_(j) and the effective information is extracted from these probabilistic models.

[0100] We then used nonparametric regression strategy for capturing the nonlinear relationships between x_(ij) and p_(ij). In many cases, this method can capture the objective relationships very well. When the data, however, contain outliers especially near the boundary on the domain {p_(ij)}, nonparametric regression models sometimes can lead to unsuitable smoothed estimates, i.e., the estimated curve exhibits some spurious waviness due to the effects of outliers. To avoid this problem, the nonparametric regression model with heterogeneous error variances as described below in Example2.

[0101] Criterion for Choosing a Graph

[0102] Once we set a graph, a statistical model (equation 8 of Example 2) based on the Bayesian network and nonparametric regression can be constructed and can be estimated by a suitable method. However, to choose the optimal graph, which gives a best approximation of the system underlying the data, it is undesirable to use the likelihood function as a model selection criterion, because the value of likelihood becomes larger in a more complicated model. Hence, a statistical approach based on the generalized or predictive error, Kullback-Leibler information, Bayes approach and so on (20) can be desirable. We therefore constructed a criterion for evaluating a graph based on our model (equation 8 of Example 2) from Bayes approach.

[0103] A criterion for evaluating the goodness of the graph can be constructed from Bayesian theoretic approach as follows:

[0104] (1) Obtain the posterior probability of the graph by the product of the prior probability of the graph π_(G) and the marginal probability of the data;

[0105] (2) Remove the standardized constant, the posterior probability of the graph is proportional to equation 9 of Example 2.

[0106] (3) Using Bayes approach, choose a graph such that π(G | X_(n)) is maximum.

[0107] (4) Determine if the computation involves a high dimensional integral (equation 9 of Example 2).

[0108] (5) If Yes to (4), use equation 11of Example 2, [see references 17 and 26 of Example 2 for integrals.

[0109] Thus, a graph is chosen such that the criterion BNRC is minimal. An advantage of using the Laplace method is that it is not necessary to consider the use of the conjugate prior distribution. Hence modeling in larger classes of distributions of the model and prior is attained.

[0110] 3. Estimating Genetic Network

[0111] Nonparametric Regression

[0112] We present methods for constructing a genetic network based on the method described in Section 2 above. The nonparametric regressor (equation 5 of Example 2) has the two components: the main effects component represented by the additive model of each parent gene, and the interactions component. In the additive model, we construct each smooth function m_(jk)(•) by B-splines (equation 9 of Example 2) (see reference18).

[0113] In the interaction terms, we use a Gaussian radial basis function. In the context of the regression modeling based on the radial basis functions, we used two methods for estimating the centers z_(jl) and the widths. This method can be termed “fully supervised learning.” An alternative method determines the values by using only the parent observation data in advance. The latter method can be employed, and a k-means clustering algorithm for constructing the basis functions can be used. Details of the radial basis functions are described further in references 7, 21 and 23 of Example 2. Hyper parameters control the amount of overlapping among basis functions.

[0114] In error variances, we consider a heterocedastic regression model and assume the structure shown in equation 6 of Example 2. The design of the constants can affect the accuracy of capturing the heteroscedasticty of the data. We set the weights according to Equation 12 of Example 2. If the hyper parameter p is set to 0, then the variances are homogeneous. If we use a large value of p, then the error variances of the data, which exist near the boundary on the domain of the parent variables, are large. Hence, if there are outliers near the boundary, this method can reduce their effects and thereby gain suitably smoothed estimates using appropriate values of p.

[0115] 4. Real Data Analysis

[0116] We demonstrated the effectiveness of the above methods through analysis of S. cerevisiae gene expression data. We observed the changes of expression level of genes on a microarray by gene disruption. By using this method we revealed gene regulatory networks between genes of Saccharomyces cerevisiae. We collected a large number of expression profiles from gene disruption experiments to evaluate genetic regulatory networks. Over 400 mutants are stocked and gene expression profiles are accumulated.

[0117] We monitored the transcriptional level of 5871 genes spotted on a microarray by a scanner. The expression profiles of over 400 disruptants were piled up in our database. The standard deviation (SD) of all genes on a microarray was evaluated and the value of SD represented roughly the error of a experiment. We needed the value of 0.5 as the critical point of the standard deviation of the expression ratio of all genes. 107 disruptants including 68 mutants where the transcription factors were disrupted were selected from 400 profiles.

[0118] We used 100 microarrays and constructed the genetic network of 521 genes from the above data. 94 transcription factors whose regulating genes are identified were found, and the profiles of 421 genes controlled by 94 factors were selected from 5871 profiles. We constructed the nonparametric regression model using 20 B-splines and 20 radial basis functions. We confirmed that the differences of the smoothed estimates against the various number of the basis functions can not visually be found, because when we use the somewhat large number of the basis functions, the hyper parameters control the smoothness of the fitted curves. We applied the two-genes effect to the interaction components. Hence, the effects of the interactions were obtained as the fitted surfaces and can be visually understandable.

[0119] We showed the roles of the hyper parameters in the prior distributions and weight constants. FIG. 1(a) of Example 2 shows the scatter plot of YGL237C and YEL071W with smoothed estimates by 3 difference values of the hyper parameters. The smoothed estimate depends on the values of the hyper parameters. FIG. 1(b) of Example 2 depicts the behavior of the BNRC criterion of the two genes in FIG. 1(a). We can choose the optimal value of the hyper parameter as the minimizers of the BNRC, and the optimal smoothed estimates (solid curve) can capture structure between these genes well. The dashed and dotted curves are near the maximum likelihood estimate and the parametric linear fit, respectively. The effect of the weight constants w1j, . . . , wnj is shown in FIG. 2(a) of Example 2. If we use a homoscedastic regression model, we obtain the dashed curve, which exhibits some spurious waviness due to the effect of the data in the upper-left corner. By adjusting the hyper parameter p in equation 12 of Example 2, the estimated curve resulted in the solid curve. The optimal value of p chosen by minimizing the BNRC criterion (see FIG. 2(b) of Example 2). When the smoothed estimate is properly obtained, the optimal value of p tends to zero.

[0120] We employed a two-step strategy for fitting the non-parametric regression model with interactions. First, we estimated the main effects represented by the additive B-spline regressors. Next, we fit the interactions components to the residuals. FIG. 3 of Example 2 depicts an example of the fitted surface to the relationship between YIL094C and its parent genes, YKL152C and YER055C. The interaction of the two parent genes leads to over expression when both parent genes increased.

[0121] In Saccharomyces cerevisiae, the GCN4 gene encodes the transcriptional activator of the “general control” system of amino acid biosynthesis, a network of at least 12 different biosynthetic pathways [reference 6 of Example 2]. Experiments showed that the consequences of the general control response upon the signal “amino acid starvation” induced by the histidine analogue 3-aminotriazole with respect to GCN4p levels. GCN4 activates transcription of more than 30 genes involved in the biosynthesis of 11 amino acids in response to amino acid starvation or impaired activity of tRNA synthetases (see reference 24 of Example 2). Purine biosynthetic genes ADE1, ADE4, ADE5,7 and ADE8 display GCN4-dependent expression in response to amino acid starvation [24]. GCN4 activates transcription of biosynthetic genes for amino acids and purines in response to either amino acid or purine starvation [24]. Those results of experiments show that there are strong relationships between purine metabolism and amino acid metabolism through GCN4. Our map of relationships fit well known relationships between purine and amino acid metabolism.

[0122] IV. Bootstrap Nonlinear Modeling of Genetic Network by Using Bayesian Network and Nonparametric Heteroscedastic Regression

[0123] In other embodiments, we modified the above approaches. A relevant feature of Bayesian network construction is in the estimation of conditional distribution of each random variable. We fit non-parametric regression models with heterogeneous error variances to the microarray gene expression data to capture nonlinear structures between genes. A problem still remained to be solved in selecting an optimal graph, which gives the best representation of the system among genes. We derived a new graph selection criterion from Bayes approach in general situation. The proposed method includes previous methods based on Bayesian networks. We also used a method for measuring the edge intensity and the degree of confidence of the direction of Bayes causality in an estimated genetic network. We demonstrated the effectiveness of the proposed method through the analysis of Saccharomyces cerevisiae gene expression data newly obtained by disrupting 100 genes.

[0124] Once we set the graph, it is desirable to evaluate its goodness or closeness to the true graph, which is unknown. For our method we needed to establish the method for measuring the edge intensity. We used a “bootstrap” method (Efron, 1979; Efron and Tibshirani, 1993) to solve this problem. By using this method, one can measure not only the intensity of edge, but also the degree of confidence of Bayes causality. To show the effectiveness of the proposed method, we analyzed gene expression data of Saccharomyces cerevisiae newly obtained by disrupting 100 genes.

[0125] 1. Bayesian Network and Nonparametric Heteroscedastic Regression for Non-Linear Modeling of a Genetic Network

[0126] 1.1 Nonlinear Baysian Network Model

[0127] As described elsewhere in this application, in the Bayesian network framework, one can consider a directed acyclic graph G and Markov assumption between nodes. The joint density function is then decomposed into the conditional density of each variable. Through formula 1 of Example 3 below, the focus of interest in statistical modeling by Bayesian networks is how one can construct the conditional densities, f_(j). One can assume that the conditional densities, f_(j), are parameterized by the parameter vectors, and information is extracted from these probabilistic models as described further in Example 3.

[0128] In certain embodiments, one can use nonparametric regression strategies for capturing nonlinear relationships between xij and Pij and suggested that there are many nonlinear relationships between genes and the linear model hardly achieves a sufficient result. In many cases, these methods can capture the objective relationships well. When the data, however, contain outliers, especially near the boundary of the domain, standard nonparametric regression models sometimes lead to unsuitable smoothed estimates, i.e., the estimated curve exhibits some spurious waviness due to the effects of the outliers. To avoid this problem, we fit a nonparametric regression model with heterogeneous error variances. If the number of the parameters in the model is much larger than the number of observations, it has a tendency toward unstable parameter estimates.

[0129] In certain cases it can be desirable to use nonparametric regression instead of linear regression, because linear regression cannot evaluate the direction of the Bayes causality or can lead to the wrong direction in many cases. We show an advantage of the new model compared with linear regression through a simple example. Suppose that we have data of gene1 and gene 2 in FIG. 1(a) of Example 3. We consider the two models gene1>gene2 and gene2>gene1, and obtain the smoothed estimates shown in Figures (b) and (c), respectively of Example 3. We then can decide that the model (b: gene1→gene2) is better than (c: gene2→gene1) by the our criterion, which is derived in a later section (the scores of the models are (b) 120.6 (c) 134.8). Since we generated this data from the true graph gene→gene2, our method yields the correct result. However, if we fit the linear regression model to this data, the mode (c) is chosen (the scores of the linear models are (b) 156.0 (c) 135.8). The method based on linear regression yields an incorrect result in this case, whereas non-linear regression analysis yields a correct result.

[0130] Even in the case where the relationship between genes is almost linear. Our method and linear regression can fit the data appropriately. However, using the linear model it is difficult to decide the direction of Bayes causality.

[0131] We have previously described the criterion BNRC. We derived the criterion, BNRC_(hetero), for choosing the graph in a general framework. By using the equation (8), the BNRC_(hetero) score of the graph can be obtained by the sum of the local scores, BNRC_(hetero). The optimal graph is chosen such that the criterion BNRC_(hetero) (equation 7 of Example 3) is minimal. An advantage of the Laplace method is that it is not necessary to consider the use of the conjugate prior distribution. Hence the modeling in the larger classes of distributions of the model and prior is attained. Genetic networks are estimated as described further in Example 3. However, regarding error variances, we consider a heteroscedastic regression model and assume the structure shown in equation 3 of Example 3.

[0132] Learning Network

[0133] In Bayesian network literature, determining the optimal network is an NP-hard problem. To resolve the networks, one can use a “greedy hill-climbing” algorithm as follows:

[0134] (1) Step 1: Make the score matrix whose (i, j)-th elements is the BNRC_(hetero) score of the graph genei→gene_(j);

[0135] (2) Step 2: For each gene, implement one of three procedures for an edge: add, remove, reverse, which gives the smallest BNRC_(hetero);

[0136] (3) Step 3: Repeat Step 2 until the BNRC_(hetero) does not reduce further; and

[0137] (4) Permute the computational order of genes and make many candidate learning orders in Step 3.

[0138] Step 4 can be desirable in situations in which a greedy hill-climbing algorithm produces many local minima and the result depends on the computational order of variables. Another problem of the learning network is that the search space of the parent genes is wide when the number of genes is large. Is such circumstances, one can restrict the set of candidate parent genes based on the score matrix, which is given by Step 1.

[0139] One also can use this learning strategy for learning genetic network and showed the effectiveness of their method by the Monte Carlo simulation method. We also checked the efficiencies of our new model through the same Monte Carlo simulations and found improvements due to the nonparametric heteroscedastic regression model, We illustrate the effectiveness of the heteroscedastic regression model in the next subsection.

[0140] Hyper Parameters

[0141] Consider the nonparametric regression model defined in equation 4 of Example3. The estimate θ_(j)is a mode of log π(θ_(j) |Xn) and can depend on the hyper parameters used. We constructed the nonparametric regression model using 20 B-splines. We confirmed that the differences of the smoothed estimates against the various number of the basis functions cannot be visually detected. When we used a somewhat large number of basis functions, the hyper parameters control the smoothness of the fitted curves. (FIG. 3(a) of Example 3 shows the scatter plot of YGL237C and YEL071W with smoothed estimates for 3 different values of the hyper parameters. The details of the data are shown in later section. The smoothed estimate strongly depended on the values of the parameters. FIG. 3(b) of Example 3 depicts the behavior of the BNRC_(hetero) criterion of the two genes in FIG. 3(a). One can choose the optimal value of the hyper parameter as the minimizer of the BNRC_(hetero) and the optimal smoothed estimate (solid curve in FIG. 3(a) can capture the structure between these genes well. The dashed and dotted curves are near the maximum likelihood estimate and the parametric linear fit, respectively.

[0142] Effects of the weight constants wij, . . . wnj is shown in FIG. 4(a) of Example 3. If one uses the momoscedastic regression model, we obtain the dashed curve, which exhibits some spurious waviness due to the effect of data in the upper-left corner. By adjusting the hyper parameter pj in equation 9 of Example 3, the estimated curve resulted in the solid curve. The optimal value of pj can be also chosen by minimizing the BNRC_(hetero) criterion see FIG. 4(b) of Example 3. When the smoothed estimate is properly obtained, the optimal value of pj tends to zero.

[0143] Finally, in Section 3 of Example 3, we provide an algorithm for estimating the smoothed curve and optimizing the hyper parameters.

[0144] Step 1: Fix the hyper parameter p_(j);

[0145] Step 2: Initialize: γ_(jk=)0, k−1, . . . , q_(j);

[0146] Step 3: Find the optimal β_(jk) by repeating Steps 3-1 and 3-2; ${{{{Step}\quad 3\text{-}1}:\quad {{Compute}:\quad {\gamma \quad {jk}}}} = {\left( {{B_{j\quad k}^{T}W_{j\quad k}\quad B_{j\quad k}} + {n\quad \beta_{j\quad k}K_{j\quad k}}} \right)^{- 1}B_{j\quad k}^{T}W_{j\quad k} \times \left( {x_{(j)} - {\sum\limits_{k^{\prime} \neq k}{\beta_{j\quad k^{\prime}}\gamma_{{jk}^{\prime}}}}} \right)}},$

[0147] for fixed β_(jk).

[0148] Step 3-2: Evaluate: Repeat Step 3-1 against the candidate value of β_(jk) and choose the optimal value of β_(jk,) which minimizes the BNRC_(hetero).

[0149] Step 4: Convergence: repeat Step 3 for k=1, . . . , q_(j), 1, . . . , qj, 1, . . . until a suitable convergence criterion is satisfied.

[0150] Step 5: Repeat Step 1 to Step 4 against the candidate value of pj, and choose the optimal value of pj, which minimizes the BNRC_(hetero).

[0151] Bootstrap Edge Intensity and Degree of Confidence of Bayes Causality

[0152] We measured the intensity of the edge and the degree of confidence of the direction of the Bayes causality in the estimated genetic network by the bootstrap method The algorithm can be expressed as follows:

[0153] (1) Make a bootstrap gene expression matrix X*_(n)={x*₁, . . . x*_(n)}^(T). by randomly sampling n times, with replacement, from the original gene expression data {x₁, . . . , x_(n)};

[0154] (2): Estimate the genetic network from X*_(n) based on the proposed method; and

[0155] (3): Repeat Step1 and Step 2 T times.

[0156] From this algorithm, we obtain T genetic networks. We define the bootstrap intensity of edge and direction of Bayes causality as follows:

[0157] Edge Intensity

[0158] If the edges gene_(i)→gene_(j) and gene_(j)→gene_(i) exist t₁ and t₂ times in the T networks, respectively, we define the bootstrap edge intensity between gene_(i) and gene_(j) as (t₁+t₂)/T.

[0159] Degree of Confidence of the Bayes Causality

[0160] If t₁>t₂, we adopt the direction gene_(i)→gene_(j) and define that the degree of confidence of causality is t_(i)/(t₁+t₂). However, we can use a certain threshold. For example, we set a real number δ and decide gene₁→gene₂ if t_(i)/t₁+t₂)>δ.

[0161] At least two ways can be used to show the resulting network. First, it can be desirable to determine the edge intensity and the degree of confidence of direction of Bayes causality in the original genetic network. One can add the intensity to each edge in the original network. From this network, one can find the reliability of the original network. Second, one can superpose the bootstrap networks and original network. However, the superposed network contains edges that have small intensities. Therefore, we can set a certain threshold value and remove the edges whose intensities are under the threshold. While setting the threshold remains a problem, it is just a visualization problem. We note that the superposed network may not hold the acyclic assumption, but much effective information are in this network.

[0162] Real Data Analysis

[0163] We illustrate the effectiveness of our methods through the analysis of Saccharomyces cerevisiae gene expression data. We monitored the transcriptional level of 5871 genes spotted on a microarray by a scanner. The expression profiles of over 400 dirsuptants were stored in our database. The standard deviation (SD) of the levels of all genes on a microarray was evaluated. The value of SD represents roughly the experimental error. In our data, we considered the value of 0.5 as the critical point of the accuracy of experiments. We evaluated the accuracy of those profiles based on the standard deviation of the expression ratio of all genes. 107 disruptants including 68 mutants where the transcription factors were disrupted could be selected from 400 profiles.

[0164] It can be appreciated that other values of SD can be considered critical for accuracy. For example, SD values can be 0.4, 0.3, 0.2, 0.1, about 0.05, about 0.01, about 0.005, about 0.001, or any other value considered to produce information having a desired reliability.

[0165] We used 100 microarrays and constructed a genetic network of 521 genes from the above data. The 94 transcription factors whose regulating genes have been clearly identified were found. The profiles of the 521 genes controlled by those 94 factors were selected from 5871 profiles.

[0166] Table 1 of Example 3 shows the gene pairs with high bootstrap edge intensities. “Inte.” and “Dire.” mean the bootstrap edge intensity and the degree of confidence of direction of Bayes causality, respectively. “F.” is the function of parent gene, i.e. “+” and “−” are respectively, induce and repress. If we cannot decide whether the function is to induce or repress, we enter “0” in “F.”. FIG. 5 of Example 3 shows the resulting partial network by superposing 100 bootstrap networks. We denote the edge intensity by the line width, and the number next to the line is the degree of confidence of the direction of Bayes causality.

[0167] From Table 1, we conclude the following: Over 60% gene pairs in the Table 1 agree with the biological knowledge. Most of the genetic sets, whose value of edge intensity equals 1, have the relation of “Related protein” in the YPD data base. Some other genetic sets, like ARO genes and PDR genes, have the same regulatory systems. These results indicated that there were some relations between genes which were previously unknown and revealed for the first time using the methods of this invention. We also notice that the other 9 genetic sets have the relation “Functional genomics”. The relation of “Functional genomics” indicated that some relations were revealed by previous information derived from large-scale, high-throughput experiments such as microarray analysis, designed to uncover properties of large groups of proteins.

[0168] Studies on the regulation of the purine biosynthesis pathway in Saccharomyces cerevisiae revealed that all the genes encoding enzyme required for AMP de novo biosynthesis are repressed at the transcriptional level by the presence of extracellular purines. Two transcriptional factors, named Bas1p and Bas2p, are required for regulated activation of the ADE genes (Daignan-Fornier and Fink, 1992) as well as some histidine biosynthesis genes (Denis et al., 1998). Those purine biosynthetic genes, like ADE1, ADE4, ADE5,7 and ADE8, display GCN4-dependent expression in response to amino acid starvation (Rolfes and Hinnebusch, 1993). The starvation for purines stimulates GCN4 translation by the same mechanism that operates in amino acid-starved cells leads to a substantial increase in HIS4 expression, one of the targets of GCN4 transcriptional activation. FIG. 5 indicates that those ADE genes and histidine biosynthesis genes are related with both BAS1 and GCN4.

[0169] A desirable feature of the methods of this invention involve the use of non-parametric heteroscedastic regression for capturing nonlinear relationships between genes and heteroscedasticity of the expression data. Therefore, our methods can be useful in the analysis of an unknown system, such as human genome, genomes from other eukaryotic organisms, prokaryotic organisms and viruses.

[0170] V. Statistical Analysis of a Small Set of Time-Ordered Gene Expression Data Using Linear Splines

[0171] Recently, the temporal responses of genes to changes in their environment has been investigated using cDNA microarray technology by measuring the gene expression levels at a small number of time points. Conventional techniques for time series analysis are not suitable for such a short series of time-ordered data. The analysis of gene expression data has therefore usually been limited to a fold-change analysis, instead of a systematic statistical approach.

[0172] We use the maximum likelihood method together with Akaike's Information Criterion to fit linear splines to a small set of time-ordered gene expression data in order to infer statistically meaningful information from the measurements. The significance of measured gene expression data is assessed using Student's t-test.

[0173] Previous gene expression measurements of the cyanobacterium Synechocystis sp. PCC6803 were reanalyzed using linear splines. The temporal response was identified of many genes that had been missed by a fold-change analysis. Based on our statistical analysis, we found that about four gene expression measurements or more are needed at each time point. These conclusions and further description can be found in Example 4 herein below.

[0174] 1. Introduction

[0175] In recent years, many cDNA microarray experiments have been performed measuring gene expression levels under different conditions. Measured gene expression data have become widely available in publicly accessible databases, such as the KEGG database (Nakao et al., 1999).

[0176] In some of these experiments, the steady-state gene express levels are measured under several environmental conditions. For instance, the expression levels of the cyanobacterium Synechocystis sp. PCC6803 and a mutant have been measured at different temperatures, leading to the identification of the gene Hik33 as a potential cold sensor in this cyanobacterium (Suzuki et al., 2001).

[0177] In other experiments, the temporal pattern of gene expression is considered by measuring gene expression levels at a number of points in time. Gene expression levels that vary periodically have for instance been measured during the cell cycle of the yeast Saccharomyces cerevisiae (Spellman et al. 1998). The gene expression levels of the same yeast species were measured during the metabolic shift from fermentation to respiration (DeRisi et al 1997). In those experiments, environmental conditions were changing slowly over time. Conversely, gene responses to an abruptly changing environment can be measured. As an example, the gene expression levels of the cyanobacterium Synechocystis sp. PCC 6803 were measured at several points in time after a sudden shift from low light to high light (Hihara, 2001).

[0178] In cDNA microarray experiments, gene expression levels are typically measured at a small number of time points. Conventional techniques of time series analysis, such as Fourier analysis or autoregressive or moving-average modeling, are not suitable for such a small number of data points. Instead, the gene expression data are often analyzed by clustering techniques or by considering the relative change in the gene expression level only. Such a “fold-change” analysis may miss significant changes in gene expression levels, while it may inadvertently attribute significance to measurements dominated by noise. In addition, a fold-change analysis may not be able to identify important features in the temporal gene expression response.

[0179] Several techniques to analyze gene expression data, such as deriving Boolean or Bayesian networks, have been used in the past (Liang et al., 1998 Akutsu et al, 2000; Friedman et al; 2000). Describing gene interactions in terms of a regulatory network is can be desirable, and developing a network model may benefit from gene expression data obtained for a large number of time points. However, data for large numbers of time points for a large number of genes is currently not available. Although the number of genes in a given organism may be on the order of several thousands, gene expression levels are often measured at only five or ten points in time.

[0180] So far, a systematic method has been lacking to statistically analyze gene expression measurements from a small number of time-ordered data. We developed a strategy based on fitting linear spline functions to time-ordered data using the maximum likelihood method and Akaike's Information Criterion (Akaike, 1971). The significance of the gene expression measurements is assessed by applying Student's t-test. This allows us to infer information from gene expression measurements while taking the statistical significance of the data into consideration. This kind of analysis can be viewed as an additional step toward building gene regulatory networks. As an example, we reanalyzed the gene expression measurements of the cyanobacterium Synechocystis sp. PCC 6803 (Hihara, 2001). Using linear splines, information can be deduced and measured data that is missed by methods using the fold-change only. By repeating our analysis with a subset of the available data, we determined how many measurements are needed at each time point in order to reliably estimate the linear spline function.

[0181] 2. Methods

[0182] Student's t-test

[0183] We assessed whether the measured gene expression rations are significantly different from unity. If for a specific gene, we can conclude that for all time points the measured expression rations are not significantly different from unity, we can eliminate that gene from further analysis. The significance level can be established by applying Student's t-test for each time point separately. Since multiple comparisons are being made for each gene, the value of the significance level a should be chosen carefully.

[0184] We define H₀ ^((i)) as the hypothesis that for a given gene the expression ratio is equal to unity at a given time point t_(i,) and H₀ as the hypothesis that for a given gene the expression rations at all time points are equal to unity. If we denote a as the significance level for rejection of hypothesis H₀, and α′ as the significance level for rejection of hypothesis H₀ ^((i)), then α′ and α are related via 1−α=(1−α′)^(a), in which a is the number of time points at which the gene expression ratio was measured. Note that by expanding the right hand side in a first order Taylor series, this equation reduces to Bonferoni's method for adjusting the significance levels (see also Anderson and Finn, 1996).

[0185] By performing Student's t-test at every time point for each gene using α′ as the significance level, we will find whether H₀ ^((i)) and therefore H₀ should be rejected. If H₀ is not rejected, we can conclude that that gene is not significantly affected by the experimental manipulation, and should therefore not be included in further analyses. If for a given gene the null hypothesis H₀ is rejected, we conclude that that gene was significantly affected by the experimental manipulation.

[0186] Analyzing Time-Ordered Data Using Linear Splines

[0187] Next, we analyzed temporal gene expression response for genes that were found to be significantly affected. The measured gene expression ratios formed a small set of time-ordered data, to which we fit a linear spline function. A linear spline function is a continuous function consisting of piecewise linear functions, which are connected to each other at knots (Friedman and Silverman, 1989; Higuchi, 1999). Whereas cubic splines are used more commonly, for the small number of data pints we can use linear spline functions more suitably. A conceptual example of a linear spline function with knots t*₁, t*₂, t*₃, t*₄, is shown in FIG. 1 of Example 4. We wish to fit a nonparametric regression model of the form x_(j)=g(t_(j))+ε_(j) to these data, in which g is a linear spline function and ε_(j) is an independent random variable with a normal distribution with zero mean and variance σ².

[0188] We estimated the linear spline function g using the maximum likelihood method. The probability distribution of one data point xj given t j, is shown in equation 3 of Example 4. The log-likelihood function for the n data points is then given by equation 4 of Example 4. The maximum likelihood estimate of the variance σ² can be found by maximizing the log-likelihood function with respect to σ². This yields equation 5 of Example 4.

[0189] The fitted model depends on the number of knots q. The number of knots can be chosen using Akaike's Information Criterion, known as the AIC (Akaike, 1971; see also Priestly, 1994). For each value of q, we calculated the value of the AIC after fitting the linear spline function as described in Example 4, and selected the value of q that yields the minimum value of the AIC. The case q=2 corresponds to linear regression. For the special case q=1, we effectively fit a flat line to the data. If we find that for a particular gene, the minimum AIC is achieved for the constant function (q=1), then we can conclude that the expression level of that gene was unaffected by the experimental manipulations. Gene expression data are typically given in terms of expression rations. At time zero, the expression ration is equal to unity by definition. This fixed point can be incorporated easily in our methodology by modifying equation 7 of Example 4. The minimum value of □² can be achieved by choosing the linear spline function shown in equation 17 of Example 4.

[0190] 3. Results

[0191] Student's t-test

[0192] We herein illustrate using Student's t-test and fitting linear spline functions, by reanalyzing measured gene expression profile of the cyanobacterium sp. PCC 6803 after a sudden exposure to high light (HL) (Hihara et al, 2001). The expression levels of 3079 ORFs were measured at zero, fifteen minutes, hone hour, six hours, and fifteen hours both for cyanobacteria exposed to HL and cyanobacteria that remained in the low light (LL) condition. Table 1 of Example 4 shows the number of measurements at each time point. Data from the cDNA expression measurements were obtained from the KEGG database (Nakao et al., 1999).

[0193] The data used for the original analysis (Hihara, 2001) may not be identical to the raw data submitted to KEGG (Hihara, personal communication). In addition, two measurement sets out of six at the t=15 minutes time point are missing in the KEGG database. Recalculating the gene expression rations from the raw data gives numbers close to the previously published results.

[0194] After subtracting the background signal intensities from the HL and LL raw data, global normalization was applied and the ration of the HL to the LL signal intensities was calculated to find the relative change in gene expression level with respect to the control (LL) condition. In the fold-change analysis, a gene was regarded as being affected by HL if its expression level changed by a factor of two or more. The statistical significance of such changes was assessed heuristically by considering the size of the standard deviation of the measurements (Hihara, 2001).

[0195] The results of the Student's t-test on the gene expression rations of each gene separately are shown in Table 2. At a significance level of α=0.001, 167 genes were found to be significantly affected by HL condition. Note that we would expect about three type-1 errors among these 167 genes. In comparison, 164 ORFs were found to be affected by the HL condition in the original analysis (Hihara, 2001).

[0196] By considering the fold-change for the psbD2 gene (slr0927), it was concluded that it was not significantly induced by HL (Hihara, 2001). This was remarkable, since this gene had been reported to be inducible by HL in the cyanobacterium Synechococcus sp. PCC 7942 (Bustos and Golden, 1992; Anandan and Golden, 1997). However, performing Student's t-test on the gene expression data for the psbD2 gene at t=6 hours yields p=3.3×10⁻⁵, suggesting that this gene was indeed affected by HL.

[0197] Analysis Using Linear Spline Functions

[0198] Next we fit linear spline functions to the measured gene expression ratios. The number of knots q were between one and five, with a fixed knot equal to unity at time zero. For q=3 and q=4, three possibilities exist for the placement of the knots between the linear segments of the linear spline. These are indicated in FIG. 2 of Example 4, together with the cases q=1, q=2, and q=5. Notice that the number of possible knot placements increases exponentially with the maximum number of knots q_(max).

[0199] In the fold-change analysis, temporal gene expression patterns were classified into six categories (Hihara, 2001), listed in Table 3 of Example 4. Fitting a linear spline function to the measured gene expression data provided a more flexible way to describe the data than categorizing. In addition, a numerical description of the gene expression response pattern is an important step in deriving a gene regulatory network.

[0200] Analyzing Expression Data Using Linear Splines

[0201] We next illustrate the usage of the AIC with an example. In the fold-change analysis, the threonine synthase gene thrC (slll688) was found to be repressed at the approximately one hour. The calculated values of the AIC for the different sets of knots are listed in Table 4. The minimum AIC was achieved for knots at 0, 15 minutes, 1 hour, and 15 hours. FIG. 3 of Example 4 shows the measured gene expression levels, together with the linear spline that was fitted to the data.

[0202] Performing this procedure for all the gene expression measures let us classify different genes based on their time-dependent response to HL. Several choices can be made as to which genes to include in this analysis. In the original analysis, genes were removed from the calculation if their expression levels were among the 2000 lowest out of 3079 ORFs (Hihara, 2001). Alternatively, we can eliminate genes if Student's t-test showed that they were not significantly affected by HL. Table 5 shows the number of genes whose measured expression levels correspond to each pattern for these different cases. For Student's t-test, a significance level q=0.001 was used.

[0203] We compared the 167 genes which were identified by Student's t-test as significantly affected by HL with the results from the fold-change analysis (Hihara, 2001), in which 164 ORFs were identified. First, we removed those genes from our analysis for which an outlier was present in the data. We define an outlier as a data point that deviates more than two standard deviations from the mean of the data at a given time point. Only one gene was found for which the measured expression data contained an outlier; the linear spline function fitted to its expression data was a flat line. None of the other gene expression levels was described by a flat line, which is consistent with the results from Student's t-test.

[0204] Next, we removed those genes whose expression level was among the lowest 2000 in order to avoid using data that are dominated by noise. The same procedure had been used for the fold-change analysis (Hihara, 2001). After removal of these genes, 107 genes remained that were significantly affected by HL.

[0205] Of the 107 genes, 42 had not been identified in the fold-change analysis (Hihara, 2001). These genes are listed in Table 6 of Example 4, together with the location of the knots that we found for each gene. For each linear spline function the percentage variance explained was calculated as a measure of the goodness of fit. As an example, FIG. 4 shows the measured gene expression ration as well as the fitted linear spline function of the gene syIR (slr0329), having four knots at zero, fifteen minutes, one hour, and fifteen hours. Of the 42 genes, the gene xyIR (slr0329) had the largest percentage variance explained (98.7%). Of the 164 ORFs that were identified in the fold-change analysis, 39 were not significantly affected by HL according to Student's t-test at a significance level q=0.001. These ORFs are listed in Table 7 of Example 4.

[0206] Finally, we established whether the number of measurements at each time point was sufficient to reliably determine the placements of knots for the linear spline function. To do so, we repeated the estimation of the linear spline function. To do so, we repeated the estimation of the linear spline function using subsets of the measured data. We then counted for how many genes the estimated knot positions changed if a subset of the data was used instead of the complete set of data. The average and standard deviation of this number for four, three, and two data points at each time point is shown in Table 8.

[0207] Even if only two data points (at the one hour time point) are removed, and four data points are used at each time point in 15% of the cases the estimated knot positions changed. Thus, in certain embodiments, four or more data points are needed for each time point to reliably deduce information from gene expression measurements.

[0208] Discussion

[0209] We have described a strategy based on maximum likelihood methods to analyze a set of time-ordered measurements. By applying Student's t-test to the measured gene expression data, we first established which of the measured genes are significantly affected by the experimental manipulation. The expression responses of those genes were then described by fitting a linear spline function. The number of knots to be used for the linear spline function was determined using Kike's Information Criterion (AIC).

[0210] Using linear spline functions permits more flexibility in describing the measured gene expression than using a nominal classification. Also, to set up a gene regulatory network, it can be desirable that the gene response as determined from gene expression measurements is available in a numerical form. Finally, positions of the knots specify those time points at which the expression of a gene changes markedly, which can be desirable in identifying its biological function. Classification of gene expression responses based on the position of the knots can be refined by creating subcategories that take the magnitude of the linear spline function at the knots into account. For instance, for linear spline functions with three knots, we may consider creating six subcategories in which changes in the gene expression level are described by (flat, increasing), (flat decreasing), (increasing, flat), (decreasing, flat), (increasing, decreasing), or (decreasing, increasing).

[0211] Applying the technique of linear spline functions to measured gene expression data, we can identify the temporal expression response pattern of genes that were significantly affected by the experimental manipulations. The response of 42 of those genes was not noticed in earlier fold-change analyses of expression data. Furthermore, it was shown that the expression response levels found in a fold-change analysis were not found to be significant by Student's t-test for 33 out of 164 genes some genes. In some embodiments, gene expression data may be nosy and are plagued by outliers. Whereas Student's t-test and maximum likelihood methods described here take the statistical significance of noisy data into account, the issue of outliers needs to be addressed separately. As a simple procedure to remove outliers, we calculated the mean and standard deviation of the data for each point in time, and removed those data that deviate more than two standard deviations or so from the mean.

[0212] Finally, the number of expression measurements needed at each time point to reliably fit a linear spline function was determined by removing some data points and fitting a linear spline function anew. It was found that if four data points per time point were used, in about 15% of the cases the knot positions will not be estimated reliably. It is therefore advisable to make more than four measurements per time point.

[0213] VI. Use of Gene Networks for Identifying and Validating Drug Targets

[0214] We describe new methods for identifying and validating drug targets by using gene networks, which are estimated from cDNA microarray gene expression data. We created novel gene disruption and drug response microarray gene expression data libraries for the purpose of drug target elucidation. We used two types of microarray gene expression data for estimating gene networks and then identifying drug targets. Estimated gene networks can be useful in understanding drug response data and this information is unattainable from clustering analysis methods, which are the standard for gene expression analysis. In the construction of gene networks in certain embodiments, we used both Boolean and Bayesian network models for different steps in analysis and capitalize on their relative strengths. We used an actual example from analysis of the Saccharomyces cerevisiae gene expression and drug response data to validate our strategy for the application of gene network information to drug discovery.

[0215] 1. Introduction

[0216] Cluster methods have become a standard tool for analyzing microarray gene expression data. However, they cannot offer information sufficient for identifying drug targets in either a theoretical or practical sense. We provide methods to determine how estimated gene networks can be used for identifying and validating target genes for the understanding and development of new therapeutics. Gene regulation pathway information is desirable for our purpose, and we used both Boolean and Bayesian network modeling methods for inferring gene networks from gene expression profiles. The procedure for identifying drug targets can be separated into two parts. At first, identify the drug-affected genes. Second, we search for the “target” genes, which are usually upstream of the drug-affected genes in the gene network. A Boolean network model is useful for identifying the drug affected genes by using “virtual gene” gene technique, described herein and a Bayesian network model can be used for exploring the druggable gene targets related to the elucidated affected genes. We applied our methods to novel Saccharomyces cerevisiae gene expression data, comprised of expression experiments from 120 gene disruptions and several dose and time responses to a drug.

[0217] 2. Gene Networks for Identifying Drug Targets

[0218] A. Clustering Methods

[0219] Clustering methods such as the hierarchical clustering and the self-organizing maps are commonly used as a standard tool for gene expression data analysis in the field of Bioinformatics. Eisen focuses on the hierarchical clustering and provides software, Cluster/TreeVew, for clustering analysis of gene expression data. De Hoon et al improved this software especially in the k-means clustering algorithm.

[0220] Clustering methods only provide the information on gene groups via the similarity of the expression patterns. However, it can be desirable to have additional hierarchical pathway information, not only cluster information to detect drug targets that are affected by an agent. We show the limitations of clustering techniques for drug targeting purposes through real data analysis.

[0221] We use two new methods for estimating gene networks from gene expression data. In this subsection, we give the brief introductions of both methods. For detailed discussions of the algorithms, please refer the papers in the references section.

[0222] B. Boolean Network

[0223] To estimate a boolean network model, we can discretize the gene expression values into two levels, 0 (not-expressed) and 1 (expressed). Suppose that u₁, . . . . u_(k) are input nodes of a node v. The state of v is determined by a Boolean functions f_(u)(ψ(u₁), . . . , ψ(u_(k))), where ψ(u₁) is the state or the expression pattern of u₁. If we have time series gene expression data, the state depends on the time t and the state of the node at time t depends on the states of inputs at time t−1. On the other hand, suppose that we have gene expression data obtained by gene disruptions. Akutsu et al. proposed the theory and methodology for estimating a Boolean network model without time delay. Maki et al. provides a system, named AIGNET, for estimating a gene network based on the Boolean network and an S-system. We use the AIGNET system for estimating Boolean network models.

[0224] Advantages of the use of the Boolean network model includes:

[0225] a) This model is simple and can be easily understood. A Boolean network model can detect the parent-child relations correctly, when the data has sufficient accuracy and information and b) the estimated Boolean network model can be directly applied to the Genome Object Net, a software tool for biopathway simulation. A disadvantage of Boolean methods is that data must be discretized into two levels, and the quantization loses information. Moreover, the threshold for discretizing is a parameter and should be chosen by a suitable criterion.

[0226] C. Bayesian Networks

[0227] A Bayesian network is a graphic representation of complex relationships of a large number of random variables. We consider the directed acyclic graph with Markov relations of nodes in the context of Bayesian network. We can thus describe complex phenomena through conditional probabilities instead of the joint probability of the random variables. Further description can be found in Example 5 herein.

[0228] Friedman et al. proposed an approach for estimating a gene network from gene expression profiles. They discretized the expression values into three values and used the multinomial distributions as the conditional distributions of the Bayesian network. However, this did not solve the problem of choosing threshold values for discretizing.

[0229] We developed a nonparametric regression model that offers a solution that does not require quantization together with new criterion, named BNRC, for selecting an optimal graph. The BNRC is defined as the approximation of the posterior probability of the graph by using the Laplace approximation for integrals. We applied the proposed method to Saccharomyces cerevisiae gene expression data and estimated a gene network. The advantages of this method are as follows: a) one can analyze the microarry data as a continuous data set, b) this model can detect not only linear structures but also nonlinear dependencies between genes and c) the proposed criterion can optimize the parameters in the model and the structure of the network automatically.

[0230] A Bayesian network has some advantageous based on the mathematics of inference, and we use the method for constructing Bayesian networks. We can construct cyclic regulations and multilevel directional models of regulatory effects from data created from logical joins of expression data from disruptants and drug response experiments by combining the Bayesian and Boolean networks in analysis. Hence, the Boolean and Bayesian networks used together can cover the shortcomings one another and we can obtain more reliable information.

[0231] 3. Applications to Microarray Data

[0232] We created two libraries of microarray data from Saccharomyces cerevisiae gene expression profiles. One was obtained by disrupting 120 genes, and the other was comprised of the response to an oral antifungal agent. (four concentrations and five time points). We selected 735 genes from the yeast genome for identifying drug targets. In YPD, 314 genes were defined as “Transcription factors”, and 98 of these have already been studied for their control mechanism. The expression profile data for 735 genes chosen for analysis included the genes controlled by these 98 “Transcription factors” from 5871 gene species measured in addition to nuclear receptor genes which have a role in gene expression regulation and are popular drug targets. We constructed network models from the data set of 735 genes over 120 gene disruption conditions. The details of the disruption data are also described in Imoto et al.

[0233] As for drug response microarray gene expression data, we incubated yeast cultures in dosages of 10, 25, 50, 100 mg of an antifungal medication in culture and took aliquots of the culture at 5 time points (0, 15, 30, 45 and 60 minutes) after addition of the agent. Here time 0 means the start point of this observation and just after exposure to the drug. We then extracted the total RNA from these experiments, labeled the RNA with cy5, hybridized them with cy3 labeled RNA from non-treated cells and applied them to full genome cDNA microarrays thus creating a data set of 20 microarrays for drug response data. In this paper, we use these 140 microarrays to elucidate drug targets using gene networks.

[0234] 4. Results

[0235] A. Clustering Analysis

[0236] In the identification of the drug targets, a popular but problematic prior art strategy has been to use clustering analysis often even with a library of base perturbation control data to compare to (( )). We have two types of microarray data, gene disruption and drug response, allowing us to compare drug response patterns to gene expression patterns caused by disruption. In the clustering analysis, if there is significant and strong similarity between the expression patterns of a single disruptant or group of disruptants and a given drug response microarray, we may conclude that an agent probably plays the same role as the disrupted gene(s). Moreover, if this disrupted gene has know functional role, we may obtain more information about the response to a drug.

[0237] Unfortunately, as if often the case with such experiments we cannot gain such a straightforward result from clustering our data. FIG. 1 shows the image representation of the correlation matrix of our microarray data. We combine the two types of data and make the matrix Z=(X:Y), where X and Y are the drug response and the gene disruption microarray data matrix, respectively. Here, each column denotes an expression pattern obtained by one microarray, and each column is standardized with a mean of 0 and variance of 1. Therefore, FIG. 1 of Example 5 depicts the information of the correlation matrix R=Z^(T)z/p, where p is the number of genes. To illustrate our methods, we focus on 735 genes for estimating gene networks and identifying drug targets.

[0238] In FIG. 1 of Example 5, the light and dark colors denote the positive and negative high correlations, respectively. Drug response microarrays have high correlation amongst each other and a low correlation with any of the gene disruption microarrays. Under such a situation, it may be difficult to identify interactions between gene disruptions and drug responses from the clustering analysis that would be meaningful in the context of drug response. We further implemented hierarchical clustering of the drug response microarrays, but this produced one cluster and we could not extract any more information on the actual drug targets from this analysis. This result was essentially unchanged when we use other distance measurements or clustering techniques. Hence, it is difficult to obtain information for identifying meaningful drug targets from the clustering methods.

[0239] B. Boolean Network Analysis

[0240] To overcome shortcomings of prior art clustering methods, we estimated gene network by using the microarry data, Z, which was created by combining gene disruption and drug response microarrays. We consider the conditions of the drug responses data as “virtual genes”, e.g., the condition 100 mg/ml and 30 min is given an assignment as the gene YEXP100mg30min. By using a Boolean network model, we found child genes of these virtual genes, with the drug affecting these child genes in progeny generational order. We focus on genes which five or more virtual genes as the parent genes, as the putative drug-affected genes, that is, genes which are under direct influence of the virtual genes (drug affect). However, a gene which has only one virtual gene as its parent gene may be the primary drug-affected gene, depending on the mode of action for a given agent. The virtual gene technique highlights the use of Boolean network models compared with the Bayesian network model in the initial screening for genes under drug-induced expression influence.

[0241] In addition, fold-change analysis can provide similar information to the proposed virtual gene technique. We identified the affected genes under certain experimental conditions by fold-change analysis. However, our virtual gene technique can improve the result of the fold-change analysis. Suppose that we find gene A and B are affected by the drug from fold-change analysis. The fold-change analysis cannot take into account the baseline interactions between geneA and geneB. That is, if there is a regulation pathway between geneA and geneB that geneA→geneB, the geneB may not be affected by the drug directly. Rather, an effect of the drug on geneA may result in an indirect effect on geneB. The virtual gene technique can take into account such interaction by using the information of the gene disruption data and thus reduce the search set to more probable target genes given available interaction data.

[0242] There is not guarantee that genes that are most affected by an agent are the genes that were “drugged” by the agent, nor is there any guarantee that the drugged target represents the most biologically available and advantageous molecular target for intervention with new drugs. Thus, even after identifying probable molecular modes of action, it is desirable to find the most druggable target genes upstream of the drug-affected genes in a regulatory network and to then screen low molecular weight compounds for drug activity on those targets. In the estimated Boolean network, virtual genes can be placed on the top of the network. Therefore, it can be difficult or sometimes impossible to find upstream information for drug-affected genes in this estimated Boolean network. In such circumstances, we use a Bayesian network model for exploring the upstream region of the drug-affected genes in an effective manner.

[0243] C. Bayesian Network Analysis

[0244] We found that a gene network can be estimated by a Bayesian network and nonparametric regression method together with BNRC optimization strategy. We use the Saccharomyces cerevisiae microarray gene expression data as described herein obtained by disrupting 120 genes. From Boolean network analysis, we found the candidate set of the drug-affected genes effectively. Druggable genes are drug targets related to these drug-affected genes, which we want to identify for the development of novel leads. We explored the druggable genes on the upstream region of the drug-affected genes in the estimated gene network by a Bayesian network method. Using a Bayesian model of network regulatory data available to us from our knockout expression library, we searched upstream of drug affected target genes, which have a known regulatory control relationship over drug affected target expression. For example, we focus on nuclear receptor genes as the druggable genes because a) nuclear receptor proteins are known to be useful drug targets and together represent over 20% of the targets for medications presently in the market. b) nuclear receptors are involved in the transcription regulatory affects that are directly measured in cDNA microarry experiments.

[0245]FIG. 3 of Example 5 shows a partial network, which includes the drug-affected genes (Bottom), the druggable genes (Top) and the intermediary genes (Middle). Of course, we can find more pathways from the druggable genes to the drug-affected genes if we admit more intermediary genes. Due to the use of the Bayesian network model, we can find the intensities of the edges and can select more reliable pathway. This is an advantage of Bayesian network models in searching for suitable druggable targets. In FIG. 3 of Example 5, druggable genes in the circle connect directly to the drug-affected genes and the other druggable genes have one intermediary gene per one druggable gene. From FIG. 3, we identified druggable genes for each drug-affected gene, e.g., we found the druggable genes for MAL33 and CDC6 shown in Table 1 of Example 5.

[0246] 5. Discussion

[0247] We describe new strategies for identifying and validating drug targets using computational models of gene networks. Boolean and Bayesian networks can be useful for estimating gene networks from microarray gene expression data. Using both methods we can obtain more reliable information than by using either method above. A Boolean network is suitable for identifying drug-affected genes by using the virtual gene technique set forth herein. Bayesian network models can provide information for upstream regions of the drug-affected genes and we can thus attain a set of candidate druggable genes. Our novel strategies are established based on the sophisticated use of a combination of two network methods. The strength of each network method can be clearly seen in this strategy and the integrated method can provide a methodological foundation for the practical application of bioinformatics techniques for gene network inference in the identification and validation of drug targets.

[0248] VII. Drug Target Discovery and Validation with Gene Regulatory Networks

[0249] Gene regulatory networks developed with full genome expression libraries from gene perturbation variant cell lines can be used to quickly and efficiently to identify the molecular mechanism of action of drugs or lead compound molecules. We developed an extensive yeast gene expression library consisting of full-genome cDNA array data for over 500 yeast strains each with a single gene disruption. Using this data, combined with dose and time course expression experiments with the oral antifungal agent Griseofulvin, we used Boolean and Bayesian network discovery techniques to determine the genes whose expression was most profoundly affected by this drug. Griseofulvin, whose exact molecular target was previously unknown, interferes with mitotic spindle formation in yeast. Our system was used to directly discover CIK1 as the primarily affected target gene in the presence of Griseofulvin. Deletion of CIK1, the primary affected target determined by-gene network discovery produces similar morphological effects on mitotic spindle formation to those of the drug. Using the base hierarchical data from the expression library and nonparametric Bayesian network modeling we were able to identify alternative ligand dependant transcription factors and other proteins upstream of CIK1 that can serve as potential alternative molecular targets to induce the same molecular response as Griseofulvin. This process for network based drug discovery can significantly decrease the time and resources necessary to make rational drug targeting decisions.

[0250] 1. Introduction

[0251] Rational drug design methodologies have previously been concentrated on optimizing small molecules against a predetermined molecular target. The randomized lead to target to phenotype screening for target selection that is currently the prevailing paradigm in drug discovery has failed to offer a more efficient and accurate target selection process even with the advent of wide scale availability of genomic information and high throughput screening processes ^((1,2,3)). The availability of genomic sequences, full genome microarrays and recent advances in gene network inference computational techniques allows for a new rational paradigm for drug target selection that takes into account global networked regulatory interactions among molecules in the genome. Accurate models of gene regulatory network data can be produced from disruptant based expression data^((4,5)) by using various computational inference techniques^((6,7)). Here we show how this gene regulatory information can be used to quickly determine the molecular networks and gene targets affected by a given compound. The same information allows for the identification and selection of alternative druggable molecular targets upstream or downstream of a drug targeted molecule in the gene expression regulatory cascade.

[0252] We have developed a gene regulatory network-driven iterative drug target discovery process. In this methodology, first large numbers of gene expression experiments are performed on single gene disruptant cell lines. This information is used to create computationally inferred maps of hierarchical gene expression control. The hierarchical regulatory information is used as a basis for evaluation of drug response experiments and for generation of hypotheses of molecular action mechanisms. Information from the literature and further biological experimentation on the elucidated regulatory subnetworks is used to understand and validate results before selecting a candidate molecule for drug targeting.

[0253] Most effective drugs in clinical use including aspirin and other popular medications have not been rationally designed to interact with a specific molecular target. Thus, even when the desired clinical effect or phenotype is achieved with these drugs, the underlying molecular mechanisms of action and thus the mechanisms of the drug's side effects remain unknown. Full genome gene expression experiments have been shown to be useful in determining alternative genes and pathways affected by a drug ^((8,9,10)), but determination of the primary molecular target for many drugs which affect hundreds of genes with standard gene expression analysis methods such as clustering is impractical without apriori information on potential targets for the drug. Here we demonstrate the use of hierarchical gene expression regulation networks from full genome expression libraries and gene network modeling techniques together with drug response expression experiments to determine the previously underlying molecular targets for the popular generic antifungal agent, Griseofulvin.

[0254] Griseofulvin is a widely prescribed oral antifungal agent that is indicated primarily for severe fungal infections of the hair and nails. While Griseofulvin's molecular action is unknown, the drug disrupts mitotic spindle structure in fungi to lead to metaphase arrest.

[0255] 2. Methods

[0256] Microarray Experiments

[0257] The yeast strain used in this study is BY4741. To monitor the gene expression profile, cells were pre-grown at 30° C. in YPD (2% polypeptone, 1% yeast extract and 2% glucose) to mid-exponential phase and were exposed with Griseofulvin by adding it to the medium to the concentrations at 0, 10, 25, 50, and 100 mg/ml. The exposed cells were harvested at 0, 15, 30, 45 and 60 min after addition by Griseofulvin and used for RNA extraction. Total RNAs were extracted by the hot-phenol method.

[0258] Boolean Network Inference Algorithms

[0259] In addition to the Boolean methods for gene disruption experiments reported elsewhere⁽¹¹⁾, we created a gene expression matrix E from a set of the drug treatment experiments combined with disruptant expression data. In the case of drug generated perturbation, we created a “virtual” gene to represent the drug affect analogous to a gene disruption. We then were able to use standard disruption matrix algorithms designed for disruption based data.

[0260] Bayesian Network Inference Algorithms

[0261] We performed non-parametric regulation and Bayesian networks to define regulatory subnetworks upstream of CIK1 using algorithms and methods reported elsewhere herein.

[0262] Data Normalization

[0263] We measured the quantities of 5871 mRNA species from 20 drug treated strains by cDNA microarray assay. A difference in fluorescent strength between Cy3, Cy5 causes bias of the expression quantity ratio. We normalized the expression quantity ratios of each expression profile. The ratio bias had a fixed trend in each spotted block, thus we calculated a linear regression to normalize the mean value ratio of each block to 1.0. The logarithm value of the ratio was used to indicate the standard expression level, therefore we found the logarithm value of ratio and calculated the average and standard deviation of these log values (see table1). The Standard Deviation (SD) of expression levels of all spotted genes from the UME6 (YDR207C) disruptant expression array for which UME6 is defined as a “Global Regulator” in YPD⁽³³⁾ disruptant was 0.4931, therefore we recognized that there are an unacceptable number of errors in array data whose overall SD was larger than 0.5 and eliminated such experiments from analysis.

[0264] Selection of Genes for Modeling

[0265] In YPD, 314 genes were defined as “transcription factors”, and 98 of these have previously been studied for control mechanism. The expression profile data of 552 genes including the genes that are controlled by these 98 “transcription factors” were selected from 5871 profiles. Thus we constructed the gene regulatory network from the expression profile data set based on the values of these 552 genes in 120 gene disruption experiments and 20 drug treatment experiments.

[0266] 3. Results

[0267] We incubated yeast cultures in dosages of 10, 50 and 100 mg in 10 ml of DMF and took aliquots of the culture at 5 time points (0, 15, 30, 45 and 60 minutes) after addition of Griseofulvin. We then extracted the total RNA from these experiments, labeled the RNA with cy5, hybridized them with cy3 labeled RNA from non-treated cells and applied them to full genome cDNA microarrays. 183 genes were downregulated by over a 2 sigma threshold among 552 genes which differed in expression between drug treated and normal yeast (FIG. 7a). Standard hierarchical clustering methodologies (FIG. 7b) applied to the combined expression libraries from drug response and gene disruption experiments, clustered genes into two major groups; first, genes affected by Griseofulvin and second, genes affected by disruption. Within the Griseofulvin clusters, genes were further grouped by dosage or time course. However, clustering did not discover any gene expression patterns that significantly indicated correlated regulation by a given gene and the drug Griseofulvin. This result would be expected except in cases where the antifungal agent affects the expression of only one discrete gene and minimal (FIG. 7b).

[0268] However, the use of gene regulatory network models, combined with the drug perturbation data allows for a hierarchical gene regulatory view of the drug's interaction with genes in the transcriptome. To generate this gene network drug perturbation data, we first created a full genome expression library of 542 single-gene disruption mutants. The 120 array data selected from the library was logically joined with the array matrices generated from the Griseofulvin experiments. A Boolean methodology designed for gene network elucidation^((11,12)) was then applied to the joint expression matrices for each time course and hierarchical regulatory maps for each experiment were generated for each dose and time point. We produced joint Boolean regulatory subnetworks for each dosage and time point experiment. The Boolean algorithm was selected for its suitability for handling joint matrices, its ability to handle looped regulatory processes and the ease creation of hierarchical directed graphs with several orders of regulatory separation. From this data we were able to identify the first order drug affects as opposed to secondary cascades initiated by those initial perturbation regulatory events.

[0269] By evaluating the Boolean data from each time and dose differential experiments we were able to identify 8 genes that were consistently and significantly suppressed as first effects at each time and drug concentration. Of these genes, CIK1 exhibited the strongest suppression effects across the experiments. CIK1 codes for a protein described in the yeast proteome database (YPD) as a coiled-coil protein of the spindle pole body involved in spindle formation and the congression (nuclear migration) step of karyogamy⁽¹³⁾. Since the action of Griseofulvin is known to affect mitotic spindle formation, which is in agreement with the function of CIK1⁽¹⁴⁾, we performed a pathological examination of a yeast strain with a disrupted CIK1 gene and yeast affected by Griseofulvin. While neither the treatment with Griseofulvin at normal physiological dosage nor the disruption of CIK1 are lethal, both cultures show similar morphological differences and growth characteristics Furthermore, microscopic examination of the mitotic spindle structure in Griseofulvin treated yeast and CIK1 deleted yeast show very similar changes of the spindle body and surrounding organizational structures ⁽¹⁵⁾.

[0270] The methodologies described here clearly demonstrate the utility of a combined expression array and computational approach using gene network techniques to rapidly ascertain and validate the molecular mechanisms of action of a given compound on a cell.

[0271] Use of such techniques will help to rationalize the target selection process of pharmaceutical development in the post-genomic era and could contribute to efficiency of discovery and a reduction in development risk for the pharmaceutical industry. The same techniques can further be applied to other biological discovery and agrochemical targeting. Our laboratories are currently replicating this discovery model in human and other biological systems. Additional descriptions can be found in U.S. Provisional Patent Application serial No. 60/395,756, filed Jul. 12, 2002 incorporated herein fully by reference.

REFERENCES

[0272] 1. Smith, A. Screening for drug discovery: The leading question. Nature. 418, 453-459 (2002).

[0273] 2. Aherne, G. W., McDonald, E. & Workman, P. Finding the needle in the haystack: why high-throughput screening is good for your health. Breast Cancer Res. 4, 148-154 (2002).

[0274] 3. Willins, D. A., Kessler, M., Walker, S. S., Reyes, G. R. & Cottarel, G. Genomics strategies for antifungal drug discovery—from gene discovery to compound screening. Curr Pharm Des. 8, 1137-1154 (2002)

[0275] 4. Hughes, T. R., et al. Functional Discovery via a Compendium of Expression Profiles. Cell. 102, 109-126 (2000)

[0276] 5. Glaever, G., et al. Functional profileing of the Saccharomyces cerevisiae genome. Nature. 418, 387-391 (2002)

[0277] 6. Friedman, N., Linial, M., Nachman, I. & Pe'er, D. Using Bayesian Networks to Analyze Expression Data. J Comput Biol. 7, 601-620 (2000).

[0278] 7. Somogyi, R. & Sniegoski, C. A. Modeling the complexity of genetic networks: Understanding multigene and pleiotropic regulation. Complexity. 1, 45-63 (1996).

[0279] 8. Marton, M. J., et al. Drug target validation and identification of secondary drug target effects using DNA microarrays. Nature Medicine. 4, 1293-1301 (1998)

[0280] 9. Heller, M. J. DNA MICROARRAY TECHNOLOGY: Devices, Systems, and Applications. Annu Rev Biomed Eng. 4, 129-153 (2002)

[0281] 10. Reynolds M A. Microarray technology GEM microarrays and drug discovery. J Ind Microbiol Biotechnol. 28, 180-185 (2002)

[0282] 11. Akutsu, T., Miyano, S. & Kuhara, S. Algorithms for identifying Boolean networks and related biological networks based on matrix multiplication and fingerprint function. J. Comp. Biol. 7, 331-343(2000)

[0283] 12. Akutsu, T., Miyano, S. & Kuhara, S. Inferring qualitative relations in genetic networks and metabolic pathways. Bioinformatics, 16, 727-734(2000)

[0284] 13. Rose, M. D. Nuclear fusion in the yeast Saccharomyces cerevisiae. Ann. Rev. Cell. Dev. Biol. 12, 663-695 (1996)

[0285] 14. Cottingham, F. R., Gheber, L., Miller, D. L., & Hoyt, A. M. Novel Roles for Saccharomyces cerevisiae Mitotic Spindle Motors. The Rockefeller University Press. 147, 335-349 (1999)

[0286] 15. Manning, D. B., Barrett, J. G., Wallace, J. A., Granok, H. & Snyder, M. Differential Regulation of the Kar3p Kinesin-ralated Protein by Two Associated Proteins, Cik1p and Vik1p. The Rockefeller University Press. 144, 1219-1233 (1999)

[0287] 16. Imoto, S., Goto, T., & Miyano, S. Estimation of genetic networks and functional structures between genes by using Bayesian networks and nonparametric regression. Pac Symp Biocomput 175-186(2002).

[0288] VII. Systems for Discovering Network Relationships and Uses Thereof In other embodiments, methods for elucidating genetic networks are provided that include Example 6 described herein. Accordingly, a desirable system can include those in which:

[0289] (1) one can collect experimental data for enabling the network to be elucidated when the genome structure is elucidated;

[0290] (2) data of all the related genes can be measured;

[0291] (3) many tools for enabling many experiments to be used, such as gene chips;

[0292] (4) an output is measured after applying a disturbance to obtain many standardized data; and

[0293] (5) analysis of the genetic relationships can be determined.

[0294] An example of such a system is presented in FIG. 1 of Example 6. FIG. 2 of Example 6 depicts schematically, a method for obtaining microarray data on expressed genes from an organism. FIG. 3 of Example 6 depicts schematically a method for assessing and quantifying the expression of a gene by comparing the amounts of gene products (e.g., RNA) from mutated cells (disruptants) in which a particular gene is disrupted with the amounts of gene products from normal (wild-type) cells.

[0295] Analysis of a large scale network can be accomplished, in some embodiments, using the guideline provided as FIG. 4 of Example 6. Time course studies can be carried out and the expression of each gene under study can be evaluated. A Boolean network model can be provided and a dynamic model of the network can be made. Positive and negative interactions can be mapped, thereby producing a gene network. A multilevel diagraph approach can be taken to relate effects of disrupting (or altering expression) each gene with respect to other genes under study.

[0296] Incorporation by Reference

[0297] All patents, patent applications and references cited in this application are incorporated herein fully by reference.

[0298] The foregoing description of embodiments of the present invention has been provided for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed. Many modifications and variations will be apparent to the practitioner skilled in the art. The embodiments were chosen and described in order to best explain the principles of the invention and its practical application, thereby enabling others skilled in the art to understand the invention and the various embodiments and with various modifications that are suited to the particular use contemplated. It is intended that the scope of the invention be defined by the following claims and their equivalence.

EXAMPLES

[0299] The following Examples are provided to illustrate embodiments of this invention. Other specific applications of the teachings in this patent application can be used without departing from the spirit of this invention. Other modifications of the methods can be used, and are considered to be within the scope of this invention.

INDUSTRIAL APPLICABILITY

[0300] Methods for determining network relationships between genes is useful in the development of lead compounds in the pharmaceutical, health care and other industries in which relationships between genes is desired. Inferential methods of this invention find application in any area of statistical analysis in which complex relationships between groups of data are desired. Such areas include engineering, economics and biology.

1 2 1 9 DNA Saccharomyces cerevisiae 1 tagccgcca 9 2 10 DNA Saccharomyces cerevisiae 2 tgggcggcta 10 

We claim:
 1. A method for constructing a gene network, comprising the steps of: (a) providing a quantitative disruptant data library for a set of genes of an organism, said library including expression results based on disruption of each gene in said set of genes, quantifying an average effect and a measure of variability of each disruption on each other of said genes; (b) creating a gene expression matrix from said library; (c) generating network relationships between said genes; and (d) determining if one or more groups of genes is expressed differently from other of said groups of genes.
 2. The method of claim 1, further comprising the step of: (e) providing a Bayesian computational model, wherein said Bayesian model comprises minimizing a BNRC criterion.
 3. The method of claim 2, wherein said step of minimizing a BNRC criterion comprises using a non-linear curve fitting method selected from the group consisting of polynomial bases, Fourier series, wavelet bases, regression spline bases and B-splines.
 4. The method of claim 1, wherein said data library is created using a drug to alter gene expression.
 5. The method of claim 2, wherein said step of minimizing said BNRC criterion further comprises selecting a Bayesian mode using a backfitting algorithm.
 6. The method of claim 2, wherein said step of minimizing a BNRC criterion further comprises using Akaike's information criterion.
 7. The method of claim 2, wherein said step of minimizing a BNRC criterion further comprising using maximum likelihood estimation.
 8. The method of claim 1, wherein said genes are associated with a cell cycle.
 9. The method of claim 2, wherein said measure of variability is variance.
 10. The method of claim 3, wherein said non-linear curve fitting method is a non-parametric method.
 11. The method of claim 10, wherein said non-parametric method for minimizing a BNRC criterion includes using heterogeneous error variances.
 12. The method of claim 11, wherein said step of minimizing a BNRC criterion further comprises the steps of: (1) making a score matrix whose (i, j)^(th) element is the BNRC^(J) _(hetero) score of the graph gene_(i)→gene_(j); (2) implementing one or more of add, remove and reverse which provides the smallest BNRC_(hetero); and (3) repeating step 2 until the BNRC_(hetero) does not reduce further.
 13. The method of claim 11, wherein said step of minimizing a BNRC criterion further comprises the step of applying a hill-climbing algorithm to minimize BNRC^((j)) _(hetero).
 14. The method of claim 11, wherein an intensity of the edge is determined using a bootstrap method.
 15. The method of claim 14, wherein said bootstrap method comprises the steps of: (1) providing a bootstrap gene expression matrix by randomly sampling a number of times, with replacement, from the original gene library expression data; (2) estimating the genetic network for gene_(i) and gene_(j); (3) repeating steps (1) and (2) T times, thereby producing T genetic networks; and (4) calculating the bootstrap edge intensity between gene_(i) and gene_(j) as (t₁+t₂)/T.
 16. A method for elucidating a gene network, comprising the steps of: (a) providing a raw data library of time-course gene expression data for a plurality of genes of an organism; (b) subtracting background signal intensities from said raw data library; (c) calculating the relative change in gene expression for each of said plurality of genes; (d) analyzing the statistical significance of said relative in gene expression using Student's t-test; and (e) fitting said changes in gene expression to a linear spline function.
 17. The method of claim 16, further comprising the step of removing from consideration, those genes whose expression levels are sufficiently low so as to be determined predominantly by noise.
 18. The method of claim 1, wherein said step of grouping comprises grouping said genes into one or more equivalence sets.
 19. A method for estimating a gene network relationship and optimizing hyper parameters of said relationship, comprising the steps of: (1) fixing hyper parameter p_(j); (2) initializing γ_(jk)=0, k−1, . . . , q_(j); (3) finding the optimal β_(jk) by repeating steps 3-1 and 3-2; ${{\left( {3\text{-}1} \right)\quad {{computing}:\quad \gamma_{jk}}} = {\left( {{B_{j\quad k}^{T}W_{j\quad k}\quad B_{j\quad k}} + {n\quad \beta_{j\quad k}K_{j\quad k}}} \right)^{- 1}B_{j\quad k}^{T}W_{j\quad k} \times \left( {x_{(j)} - {\sum\limits_{k^{\prime} \neq k}{\beta_{j\quad k^{\prime}}\gamma_{{jk}^{\prime}}}}} \right)}},$

for fixed β_(jk); (3-2) repeating step 3-1 against a candidate value of β_(jk) and choosing the optimal value of β_(jk), which minimizes BNRC_(hetero) (4) repeat step 3 for k=1, . . . , qj, 1, . . . , qj, 1, . . . until a suitable convergence criterion is satisfied; and (5) repeat steps 1 to Step 4 against a candidate value of pj, and choosing the optimal value of pj, which minimizes the BNRC_(hetero).
 20. A method for constructing a gene network model of a system containing a network of genes comprising using a Bayesian computational model, wherein said Bayesian computational model comprises minimizing a BNRC criterion.
 21. The method of claim 20, wherein minimizing the BNRC criterion comprises using a non-linear curve fitting method selected from the group consisting of polynomial bases, Fourier series, wavelet bases, regression spline bases and B-splines.
 22. The method of claim 20, wherein minimizing the BNRC criterion comprises selecting a Bayesian mode using a backfitting algorithm.
 23. The method of claim 20, wherein minimizing the BNRC criterion comprises using Akaike's information criterion.
 24. The method of claim 20, wherein minimizing the BNRC criterion comprises using maximum likelihood estimation.
 25. The method of claim 20, wherein minimizing the BNRC criterion comprises using a non-linear curve fitting method, wherein the non-linear curve fitting method is a non-parametric method.
 26. The method of claim 25, wherein the non-parametric method includes using heterogeneous error variances.
 27. The method of claim 26, wherein minimizing the BNRC criterion further comprises the steps of: (1) making a score matrix whose (i, j)^(th) element is the BNRC^(j) _(hetero) score of the graph gene_(i)→gene_(j); (2) implementing one or more of add, remove and reverse which provides the smallest BNRC_(hetero); and (3) repeating step 2 until the BNRC_(hetero) does not reduce further.
 28. The method of claim 26, wherein minimizing the BNRC criterion further comprises the step of applying a hill-climbing algorithm to minimize BNRC^(j) _(hetero).
 29. The method of claim 26, wherein an intensity of the edge is determined using a bootstrap method.
 30. The method of claim 29, wherein said bootstrap method comprises the steps of: (1) providing a bootstrap gene expression matrix by randomly sampling a number of times, with replacement, from the original gene library expression data; (2) estimating the genetic network for gene_(i) and gene_(j); (3) repeating steps (1) and (2) T times, thereby producing T genetic networks; and (1) calculating the bootstrap edge intensity between gene_(i) and gene_(j) as (t₁+t₂)/T.
 31. The method of claim 20, wherein the Bayesian computational model is used to analyze a gene expression profile of the system.
 32. The method of claim 31, wherein the gene expression profile comprises the level of gene expression of each gene in the system.
 33. The method of claim 32, wherein at least one gene in the system is disrupted.
 34. The method of claim 32, wherein the gene expression profile comprises a sub-gene expression profile and wherein the sub-gene expression profile comprises the level of gene expression of each gene in the system when at least one gene is disrupted in the system.
 35. The method of claim 34, wherein the gene expression profile comprises at least two different sub-gene expression profiles.
 36. The method of claim 32, wherein the system is treated with an agent.
 37. A method for constructing a gene network model of a system containing a network of genes comprising using a Bayesian computational model and a Boolean method.
 38. The method of claim 37, wherein said Bayesian computational model comprises minimizing a BNRC criterion.
 39. The method of claim 37, wherein the Bayesian computational model and the Boolean method are used to analyze a gene expression profile of the system.
 40. The method of claim 39, wherein the gene expression profile comprises the level of gene expression of each gene in the system.
 41. The method of claim 40, wherein at least one gene in the system is disrupted.
 42. The method of claim 40, wherein the gene expression profile comprises a sub-gene expression profile and wherein the sub-gene expression profile comprises the level of gene expression of each gene in the system when at least one gene is disrupted in the system.
 43. The method of claim 42, wherein the gene expression profile comprises at least two different sub-gene expression profiles.
 44. The method of claim 40, wherein the system is treated with an agent.
 45. A data file comprising a gene network model constructed by the method of claim
 20. 46. The data file of claim 45 in a computer readable form.
 47. The data file of claim 45 accessible from a remote location.
 48. The data file of claim 45 accessible from an internet web location.
 49. A method for identifying a target gene of an agent in a system containing a gene network comprising: (a) constructing a first and second gene network model using a Bayesian computational model, wherein said Bayesian computational model comprises minimizing a BNRC criterion, wherein the first gene network model is obtained by analyzing a first gene expression profile and the second gene network model is obtained by analyzing a second gene expression profile, wherein the first gene expression profile is obtained from the system without being treated with the agent and the second gene expression profile is obtained from the system being treated with the agent, and (b) analyzing the first and second gene network model using said Bayesian computational model, wherein the agent is considered as a gene in the system, and wherein a target gene of the agent is identified.
 50. The method of claim 49, wherein the target gene is a gene directly affected by the agent.
 51. The method of claim 49, wherein the target gene is a gene indirectly affected by the agent.
 52. A data file containing the identity of one or more target genes of the agent obtained according to the method of claim
 49. 53. The data file of claim 52 in a computer readable form.
 54. The data file of claim 52 accessible from a remote location.
 55. The data file of claim 52 accessible from an internet web location.
 56. A method of providing a service comprising receiving an agent from a party, and identifying a target gene of the agent for the party according to the method of claim
 49. 57. The method of claim 56, wherein receiving an agent comprises receiving the identity of the agent.
 58. A method of providing a service comprising receiving an agent from a party, and identifying a target gene of the agent for the party using the gene network model constructed according to the method of claim
 20. 